If Louis XIV belongs on the list of counterexamples to the OP's claim, then not only does Louis XIII, as has been pointed out by others, but surely also:

Philip II Augustus (acceded age 15, though no regency--in fact he was crowned at 14 a year before his father's death and basically served as regent for his ailing father, which doubtless eased the transition) who is called "Augustus" for basically creating the state that Louis XIV identified with himself, who brought the English to their knees and invaded (without which invasion the English monarchy would likely have been able to shelve Magna Carta), and broke the Welfs at Bouvines; and

his grandson Saint Louis IX (acceded age 12, under the Regency of his mother Blanche of Castile), the "arbiter of Europe", a great patron of the arts, co-patron of the Franciscan Order, after whom the French call the 13th century "the golden century of Saint Louis". (Incidentally Louis IX is the most recent common patrilineal ancestor of all subsequent French kings.)

What about Alexander the Great?

Succeeded his father to the throne at the age of 16* 20.

As others stated, it's hard to tell what is "good at the job", but I think it's also hard to say alexander was a "failure at the job".

*Edit: as @LangLangC noted, I provided incorrect information. Alexander the Great was succeeded his father to the throne at the age of 20. I guess it's not count as child anymore and doesn't answer the question.

Gustav II Adolf of Sweden comes to mind, but may not meet your definition of becoming king as a child. Crowned at the age of 16, officially took the throne at 17, his leadership made Sweden one of the major powers of the Thirty Years War and set the stage for the advance of the Swedish Empire.

He is generally regarded as one of the greatest military leaders of his time, and his innovative integration of all the aspects of warfare at the time (infantry, cavalry, artillery, and logistics) has earned him recognition as the 'Father of Modern Warfare' among military scholars. His involvement in the First Battle of Breitenfeld was a significant contributing factor to what would become the first major Protestant victory of the Thirty Years War.

He also made significant political reforms that enabled the advancement of the Swedish Empire, including changes to the census system that allowed for more efficient taxation and conscription, and numerous other changes that transformed the Swedish economy and culture from something resembling the late medieval era to a modern (well, modern by mid 17th century standards) form over the course of only a few decades.

Although not a King or Queen, Tenzin Gyatso (born Lhamo Thondup), the 14th Dalai Lama, seems to have done reasonably well. He was formally enthroned in 1940, at the age of 4, and took up his full range of duties in 1950, at the age of 15.

IMHO the only way for you to get an answer to your question to look at lists of rulers who came to the throne as children and try to see what their historical reputations are.

In Scotland the monarchs who inherited as children or teenagers were:

Malcolm IV "The Maiden" 1141-1165, became king 1153 aged 12.

Alexander II 1198-1249, became king 1214 aged 16.

Alexander III 1241-1286, became king 1249 age 7.

Margaret "The Maid of Norway" 1183-1290, became queen 1286 age 3.

David II Bruce 1324-1371, became king 1329 age 5.

James I Stewart 1394-1437, became king 1406 age 11.

James II Stewart 1430-1460, became king 1437 age 6.

James III Stewart 1451-1488, became king 1460 age 9.

James IV Stewart 1473-1513, became king 1488 age 15.

James V Stewart 1512-1542, became king 1513 age 1.

Mary Stuart 1542-1587, became queen 1542 age 6 days.

James VI Stewart 1566-1625, became king 1566 age 1.

The more or less official foundation of the kingdom of England was in 927.

Monarchs who came to the throne as children and teenagers were:

Edmund I c. 921-946, became king 939 age about 18.

Eadwig c. 940-959, became king 955 aged about 15.

Edgar the Peaceful c. 943-975, became king 959 aged about 16.

St. Edward the Martyr c. 962-978, became king aged about 13.

Aethelred II the Unready c.968-1016, became king 978 age about 9 or 10, ready or not.

Edgar II Aetheling c. 1051-c.1126, the rightful heir of the Anglo-Saxon kings, he was chosen king in October 1066 aged about 15 but submitted to William the Conqueror in December.

Henry III 1207-1272, became king 1216 age 9.

Edward III 1312-1377, became king 1327 age 14.

Richard II 1367-1400, became king 1377 age 10.

Henry VI 1421-1471, became king the first time age 9 months.

Edward IV 1442-1483, became king the first time age 18.

Edward V 1470-1483?, became king age 12.

Henry VIII Tudor 1491-1547, became king 1509 age 17.

Edward VI 1537-1553, became king 1547 age 9.

Jane 1536/37-1554, became Queen 1553 age about 15 or 16.

Theses two lists may be a big enough sample to decide whether there are any particular trends among monarchs who begin their reigns as children.

Here is another list, monarchs of France from 843.

Louis III (863/65-882, became king 879 aged 13 to 16.

Carloman II (c. 866-884), became joint king age about 12 or 13, sole king age 15 or 16.

Charles III the Simple (879-929), became rival king 893 age 13 or 14, sole king 898 age 18 or 19, deposed in 922.

Louis IV From Overseas (920/21-954), became king 936 age about 15.

Lothair (941-986) became king 954 age 13.

Philip I (1052-1108), became king 1060 age 8.

Louis VII (1120-1180) The young, became king 1137 age 17.

Philip II (1165-1223) Augustus, became king 1180 age 15.

Louis IX (1214-1270), became king 1226 age 12.

Philip IV (1268-1314), became king 1285 age 17.

John I (1316) lived and reigned for 5 days.

Charles VI (1368-1422), became king 1380 age 11.

Charles VIII (1470-1498), became king age 13.

Francis II (1544-1560), became king 1559. age 15.

Charles IX (1550-1574), became king 1560 age 10.

Louis XIII (1603-1643) became king 1610 age 8.

Louis XIV (1638-1715), became king 1643 age 4.

Louis XV (1710-1774), became king 1715 age 5.

Louis XVII (1785-1795), became king in the eyes of royalists 1793 age 7.

Napoleon II (1811-1832), became titular emperor from 4 to 6 April, 1814, age 3, and from 22 June to 7 July 1815 age 4.

Henry V (1820-1883), became king 2 August to 9 August 1830 age 9.

Or you could look at lists of the best and the worst monarchs in history and see what what ages they became monarchs.

I believe I will focus on the Eastern Kings and Emperors since the other answers provide ample information on the Western side.

Shapur II of Persia:

When his father Hormazd II died, he was still in his mother, the Empress Dowager Ifra Hormizd's womb. His elder brother Prince Adur Narseh inherited the throne.

As a result of revolts and conspiracy by nobles against cruelty of the Emperor, Shah Adur Narseh was murdered, his second brother was blinded, third was imprisoned but escaped to Roman Empire. Bereft of any other option, Lords of Persia turned to the unborn Shapur II.

The crown of Persian Empire was placed on his pregnant mother's pudenda as per the legend and throne was reserved for the unborn child. (That is disputed by some historians given that sex of the child could not be known and if it was a girl, she couldn't take the throne). So he is presumably the only monarch in the world who was crowned in utero and was born an Emperor unlike other Princes.

Until he was deemed to have come of age (16 years), Nobles effectively ran the Empire in capacity of regents.

The unborn child would go on to rule for seventy years, longest reign of any Persian Emperor and become one of the Greatest Emperors of Persia. His reign is considered the most illustrious in Persian history.

He did however adopt a policy of persecuting Christians when Constantine the Great of Roman Empire converted to Christianity, fearing that Christians might take over the Persian Empire too. He was neverthless generous towards the Jews and other minorities.

Mehmed II the Conqueror of Ottoman Empire:

Mehmed II, famously known as the Conqueror or Fatih Sultan, first ascended to the throne at the age of 12 when his father Murad II abdicated the throne and retired to spend the rest of his days in Anatolian countryside.

Sensing potential weakness in the Ottoman Empire with a boy at helm, The Western European powers launched another Crusade, as the Papal representative Cardinal Julian Cesarini had convinced the European powers that breaking a truce made with Muslims was not a sin. Mehmet called his father back from the retirement who refused. Upon which, Mehmet penned another angry letter saying:

If you're the Sultan, come and lead your armies. If I am the Sultan, I command you to come and lead my armies.

At this, Murad II returned and defeated the European powers at Varna. After that, Murad II assumed the throne again and reigned till his death. Thus began the second reign of Mehmet II which saw rapid expansion of Ottoman Empire and ultimate defeat of the Eastern Roman Empire.

Akbar the Great of Mughal Empire

Akbar was the third Mughal Emperor, grandson to Babur and son to Humayun and a scion of the House of Timur (Patrilineally) and Genghis Khan (Matrilineally).

He was born in exile when his father was deposed by the Usurper Sher Shah Suri. After the usurper died and chaos followed the succession of his son Islam Shah Suri, the deposed Emperor Humayun took advantage of the situation and reconquered Delhi with the aid of Persian Shah Tahmasp I. A few months later Emperor Humayun died and 14 years old Akbar ascended to the throne. He was however deemed too young to rule as yet and his tutor, Bairam Khan ruled as regent until he came of age.

Akbar's reign saw absolute domination of the Mughal Empire over India and the young Emperor was never beaten in field. He ultimately became one of the three or four people who managed to bring most of Indian subcontinent under one Empire. His achievements were not just military, he encouraged scientific innovations and their application in warfare, created a huge library including Sanskirit, Persian, Greek, Arabic and Latin works, formed legal and cultural foundations of Mughal Empire, won the loyalty of his majority non-Muslim subjects through respect, tolerance, abolishment of jizya, equal employment opportunities, took the Indian economy to new heights.

He reigned for nearly half a century and is without a doubt the greatest Emperor House of Timur ever produced.

Al-Hakim Bi Amr Allah of Fatimid Caliphate:

Now this here is a controversial entry. If there ever was a man who was equal parts good and equal parts evil, that would be him.

He was born in Cairo to Caliph Al-Aziz. When he was 11, his father fell sick on return trip from Syria where he had been to visit the Frontlines against the Eastern Roman Empire. His father passed away within hours of seeing his heir for the last time and 11 years old Prince Abu Ali Mansur was proclaimed "By the grace of God, Caliph of the Prophet, Commander of the Faithful and Sultan of Egypt, Syria and Maghreb" by the court and took the regnal name Al-Hakim bi-Amr Allah (Meaning Ruler by the decree of God).

As long as the Caliph was underage, his tutor Barjawan, the Qadi and Berber leader Ibn Ammar were going to act as a regency council as per his father's will. But the first challenge soon presented itself when Berbers demanded concessions and offices from the young Caliph, who obliged them. Their leader Ibn Ammar effectively took the role of the sole regent as the first Wastija (Intermediary between the Caliph and Administration) and tyranny ensued. Barjawan soon attracted Ibn Ammar's rivals among the Berber military leaders and successfully deposed him. As long as Barjawan lived, he maintained a balance between all the military and civil factions in the Caliph's name as regent.

Following Barjawan's murder, the 15 years old Caliph took personal command of the government and began a purge of the Fatimid elites which saw wholesale murders of Ibn Ammar and other troublesome yet powerful Berber leaders. He also limited the number of years one could remain in the office of Vizier or Wastija to curb the power of nobles and increase his own powers.

Now he had brought the situation under control and the Fatimids started expanding again. But that only created more problems. Baghdad Manifesto happened. It was a declaration made by Twelver-Shia and Sunni genealogists who testified on behest of Sunni Abbasid Caliph Al-Qadir (Rivals of Fatimids) that Fatimids were in fact not descended from the sacred line of Ali and Fatimah, they were in fact descended from a Jew. Al-Qadir had hoped to halt the Fatimid spiritual and military expansion via this declaration and it harmed Fatimid prestige considerably.

Even on the home front, tensions started simmering again. Turk and Berber armies were once again at each other's throats, Sunnis started launching attacks on Ismaili communities in North Africa. However the Caliph managed to keep the armies from an outright civil war and did not lose any lands in the West. To counter the growing enemies, Al-Hakim launched a fierce diplomatic campaign with which he cultivated friendly ties with powerful Eastern Roman Empire and even as far as the Song Emperors of China.

Other than facing his foes both on home front and abroad competitively, he is also infamous for persecution of Christians, Jews and ruthlessness he showed to all his foes in general. He is also famous for making Cairo the world's centre of culture and learning, with most important contributions being building the famous "House of Knowledge" and putting extra emphasis on education of his subjects.

He died young at the age of 36 in possible assassination. He went out for his regular journeys to the hills outside Cairo and never came back. A search turned up only blood stained clothing. He had become obsessed with asceticism before that so it is also possible that he may have left the Kingdom of his own accord.

Some people hated him and called him mad, some people loved him to the point that they considered him a living God (Not figuratively, literally).

One successful child ruler from Israel was King Josiah of Judah, who took the throne at the age of 8. He succeeded his father Amon, who was assassinated after reigning for only two years. Josiah reigned 31 years, and completely reformed the religious system in Israel according to the Law of God. The book of 2 Kings records about him “And he did that which was right in the sight of the LORD, and walked in all the way of David his father, and turned not aside to the right hand or to the left.” (Note: King David was his distant ancestor, and often the king by which others were judged.) The text doesn’t describe any contest to his reign, nor any opposition to leading Israel back to God.

Josiah died while opposing King Necho II of Egypt.

Well, this is somewhat of a technicality, but Gustavus Adolphus of Sweden was under a regency for a few months after his father had died, before being declared of age at 17 years old. Even if there are those who have questioned his decision to enter the Thirty Year's War, there is no doubt that he brought Sweden from being on the brink of military catastrophe to one of the most powerful nations in Europe, that he modernized government and helped revolutionize military tactics, or that the entry in the War was a permanent turning point for the protestant cause.

As far as I know, there were no rebellions during his reign; he even managed to have good working relationships with the sons of nobles that had been executed by his father. After his death in battle, he was granted the epithet "the Great" by the Swedish estates (though it is almost never used). For a long time, he was perhaps the main figure in Swedish nationalism, and almost a sort of protestant saint.

Less of a technicality, Charles XI of Sweden later inherited the Empire that Gustavus Adolphus had helped create. After being embroiled in the Franco-Dutch war, which nearly lost him large parts of the German and former Danish possessions, he set out to reform the realm, centralizing power to himself, reducing the lands of the nobles, strengthening the military, and attempting to unify the laws. Despite harsh treatment of the population of the former Danish and Norwegian provinces, he did succeed in incorporating them in Sweden, politically and culturally.

While less admired than some other Swedish kings, Charles XI:s reign saw the longest peace for over 150 years, and while his reforms ultimately proved to have been less than what was needed, they did give Sweden a fighting chance in the Great Northern War, which broke out after his death. Charles also enjoyed a place in legend as an unusually just king, who would travel the land in disguise to right wrongs.

Sources

For a general work on Sweden's 17th century, I'd recommend the fourth volume of Sveriges Historia. For the life of Gustavus Adolphus, Sverker Oredsson's Gustav II Adolf is useful. For Charles XI, Göran Rystads Karl XI is good.

A hands-down yes for the Kangxi Emperor of Qing-dynasty China. He became nominal Emperor at age 7 and had a 6-year regency; then ruled directly from age 13 to his death in his late 60s.

His reign was considered one of the most prosperous and productive periods in Chinese history, and (unlike the Louis XIV example) it unquestionably extended to future generations, with his grandson the Qianlong Emperor having an equally long and prosperous rule.

Kangxi himself was enthroned on only the 17th year of the Qing dynasty (a tumultuous 17 years, since he was the 4th Qing emperor). His early achievements include consolidating that dynasty and suppressing the remaining Ming loyalists.

In addition to military success, over the span of his rule he doubled the contents of the national treasury while also managing to lower taxation, sponsored creation of a tremendously important comprehensive dictionary of Chinese (among other scholarly patronage), absorbed (through Jesuit contacts) Western knowledge without falling prey to colonialism, and dramatically reorganized the government to reduce corruption by intervening-level officials.

While among Qing royalty Qianlong might be his superior in influence and importance, Kangxi laid the foundations for the best 150 years of the dynasty and was indisputably an exceptional ruler.

A child monarch generally has to have a Regency period, where someone else actually wields all the powers of state in their behalf, until they reach adulthood and can reasonably be expected to do it themselves. The thing about regencies is that, historically, they tend to have a pretty bad track record keeping their charge alive.

A regent (essentially acting King, and thus above the law) already has little incentive to hand over all that sweet, sweet power. Adding to the equation an heir with a strong personality (and thus likely to completely take over from their regents immediately upon majority), and the incentive drops to near zero. So natural selection would favor weak rulers surviving their regencies.

I can't find a good source for child monarchs/regencies, but a quick proxy would be to begin with the list of longest serving monarchs (on the grounds that (a) it is much more likely that a long serving monarch will have started earlier in life, and (b) permitting the ruler to endure on the throne indicates that his contemporaries felt his rule was "good".)

• Sobhuza of Swaziland has a notable absence of derogatory comments, but is modern. OP didn't limit the question to pre-modern, but I think it should be; the challenges of a modern monarch are distinctly different from the pre-modern monarch.

• Bernard of Lieppe, known as "the Bellicose". That nickname would probably qualify him as "good" to his contemporaries, but "bad" to our contemporaries.

• William IV Princely count of Henneberg-Schleusingen qualifies on the second ground above - he inherited at 5 years old and reigned for 80 years. For 80 years his subjects couldn't find a better alternative, so he must have been a good ruler.

• K'inich Janaab Pakal of Mexico

  During a reign of 68 years, the longest known regnal period in the history of the Americas, the 30th longest worldwide and longest until Frederick III in the 15th century, Pakal was responsible for the construction or extension of some of Palenque's most notable surviving inscriptions and monumental architecture.

It might also be useful to consider the full list of regencies.

There were a couple of notable pharaohs:

• Thutmose III, born 1481 BC, reigned 1479–1425 BC, was only about 2 year old when he became pharaoh. His co-regent was his stepmother Hatshepsut, but she continued as co-ruler / pharaoh until her death when Thutmose was 22 years old. According to Wikipedia, Thutmose is "Widely considered a military genius by historians".
• Amenhotep III, born about 1388 BC, regnal dates are disputed but he was probably around 6 to 12 years old when he became pharaoh. His reign "was a period of unprecedented prosperity and artistic splendour, when Egypt reached the peak of its artistic and international power."

In Iran,

• Shapur II, born 309 AD, reigned 309-379 AD, was an infant who became Iran's longest reigning monarch. "His reign saw the military resurgence of the country, and the expansion of its territory, which marked the start of the first Sassanian golden era."

For Pontus,

• Mithridates VI, born 134 BC, reigned 120–63 BC, was 14 years old. "Mithridates is remembered as one of the Roman Republic’s most formidable and successful enemies,"

In Europe,

• David II, born 1324, King of Scotland from 1329 to 1371, was 5 years old. His reign had its ups and downs but "By the time of his death, the Scottish monarchy was stronger, and the kingdom and royal finances more prosperous than might have seemed possible."
• Edward III, born 1312, reigned 1327-77, was 14 years old when he became king with Roger Mortimer as the effective ruler. In 1330, the young king seized power and had Mortimer executed. "Edward III transformed the Kingdom of England into one of the most formidable military powers in Europe. His long reign of 50 years was the second longest in medieval England and saw vital developments in legislation and government"

Peter the Great of Russia took the throne at age 10, and Ivan the Terrible (who was terrifying, not incompetent) became Prince of Moscow at age 3. Both of them were successful at centralizing power, modernizing the country and conquering its neighbors. Ivan did a lot of other things that we’d consider bad today, and Peter survived several power struggles in his childhood.

In contrast, the worst thing you could say about Wilhelmina of the Netherlands, who became Queen at age 10, is that she was forced into exile by the Nazi occupation, which I suppose is a “crisis of legitimacy.” Of course, a modern constitutional monarch before the era of tabloids, much less social media, had a much easier job than an absolute monarch.

As Mark C. Wallace's comment points, the answer depends on what you see as a good king. However, there are kings who became kings while children and are still regarded as great kings - at least, among the most famous in their countries.

Two examples:

Louis XIV of France: king at 4 years old, declared to have reached the age of majority (and regency ended) at 13 years old.

James I of Aragon: king at 5 years old. Still the most important king of Aragon - at least from a Catalan perspective.

Edit to address the comment that Louis XIV was not so good:

If Louis XIV wasn't good at his job because he inherited the most powerful richest and most populous kingdom in Europe and left it in way poorer state (which is a good point), then credit should go to the previous king, his father Louis XIII who would then make a better answer to the question, since he also became king as a child (at 9 years old).