If "the Kingdom of God comes not with observation", how can it be "taken by violence"?

"Violence" may not be the best translation of the word, because, as you see in your case, it can have unintended inferences. "Force" would be a better rendering. You can see that alternate meaning of the Greek word here: https://www.blueletterbible.org/lang/lexicon/lexicon.cfm?Strongs=G971&t=KJV

How do you take the kingdom by force? The kingdom of God is the rule of God over you, personally -- over your passions, ego, strength, thinking, words, actions, plans, and so on. When Jesus rules over these aspects of our lives, then we live in a way that is pleasing to Him. You can see the inner Kingdom of God life in Matthew chapters 5-7. So you know yourself in the kingdom when you read the Bible and follow the direction given -- especially when it takes you in directions you would never have thought of on your own. When you obey it, then it can be said that Jesus rules over you in that case.

There is no way to get this kingdom character of obedience except to fight against personal sinful tendencies and bad habits and acknowledge the uncomfortable lies we've lived with for so long.

Due to the sin nature within, we don't give up doing things on our own. That's where the force comes in. We want to be kings and we won't give up our thrones readily. Paul the apostle expressed that force as "reckoning yourself dead to sin" in Romans 6:11, and not obeying the sin inside (Romans 6:12), which means we must use force against those passions and habits to keep them at bay. Those who are Christ's -- those who are ruled by Jesus and not sin -- have "crucified" the fleshly passions (Galatians 5:24).

In 1 Corinthians 9, you'll see a pattern of self-denial in Paul; he cut off his right to do things he would have done, but chose to go the hard way to present the gospel in a clear manner -- by living it.

In 1 Corinthians 9:24-27, Paul says he disciplines himself lest he should be disqualified from receiving the prize at the judgment seat, then in the very next chapter he goes over the sad end of those who did not discipline themselves. The kingdom is taken by violence in the sense that those who have a strong desire to overcome their personal issues will strive to enter it. Armchair church-goers need not apply.

So don't take the word "violence" as someone who is crazy, yelling, and brandishing weapons out of control. That meaning doesn't fit the context. Olympic athletes must apply "force" (self-discipline) against their own bad habits and the pull of culture that they may become the best they can be, and they are not crazy out of control.

"The violent take it by force" - this simply means you need to strive to enter the Kingdom of God.

Jacob fought violently the whole night and when he understood who he was fighting with, a new warfare began. Jacob would not let go the leg of the Angel - no matter how the Angel asked/pled with him. This is spiritual violence. Jacob "fought with God and man and won" - he was given the name Israel "As a prince thou hast fought and won". Jacob was violent and he got the kingdom of heaven by force.

Speaking along the same line:

Heb 12:4 In striving against sin, ye have not yet resisted unto bloodshed.

We should strive violently against sin even to the point of blood shed. This is spiritual warfare.

2 Corinthians 10:4-6 For the weapons of our warfare are not carnal, but mighty through God for the pulling down of strongholds, casting down imaginations and every high thing that exalteth itself against the knowledge of God, and bringing into captivity every thought to the obedience of Christ, and being in readiness to avenge all disobedience when your obedience is fulfilled.

We have so many examples in the life of Christ:

1. The woman with 12 years of bleeding - Violently strived against the crowd and finally succeeded in touching the corner of Jesus' coat

2. The friends of the man with palsy - Violently broke open the roof and lowered him and succeeded in getting forgiven and healed

3. The Syrophonecian lady - violently pled with Jesus and would not leave unless her daughter was healed

4. Tle leper - "If Thou wouldest, thou canst make me clean" - Violently struggled throught the crowd to get healed

5. The blind men on the way to Jericho - "Jesus, thou Son of David, have mercy on me" - violently shouted till they were heard

Jesus said "Luke 13:24 “Strive to enter in at the strait gate, for many, I say unto you, will seek to enter in and shall not be able."

You should violently strive if you would enter heaven. Only the violent can take it. If we are complacent in this struggle, we would loose. We must not let go of Jesus until we have received what we want - I am not talking about temporal blessings, but spiritual development. We must not let go until we are assured of the forgiveness of our sins, of the victory over the sin with which we are struggling, of the love of God filling our hearts - we must struggle like Jacob and only then will God give the desires of our heart.

The verses cited, when taken in the broader context of the pericopes to which they belong, both emphasize the importance of believers taking an active role in preparing themselves for the Kingdom of Heaven.

What Matthew reports Jesus to say in 11:12 relates to what He had just said to the multitude:

Matthew 11:11

Verily I say unto you, among them that are born of women, there has not arisen a greater than John the Baptist. Notwithstanding, He that is less, in the kingdom of Heaven is greater than he.

The eastern Church Fathers (e.g. John Chrysostom, 4th century) understood Christ to be referring to Himself in Matthew 11:11. He, Christ, was not born of woman - i.e. a married woman, gyne in Greek - but of a virgin (parthenos). He, Christ, is "less" than John the Baptist in years (the Greek microteros also means "younger"), but He is greater in the Kingdom in Heaven. The Kingdom of Heaven suffering violence and the violent taking it by force refers to what His followers must do to attain the Kingdom of Heaven: leave one's father and mother (Matthew 19:29, Mark 10:29, Luke 18:29); despise one's own life (John 12:25); etc. (See Chrysostom's Homily XXXVII on Matthew).

Similarly, I think we must consider Luke 17:20 in the context of the verse that follows:

Luke 17:20–21 (KJV 1900)

And when he was demanded of the Pharisees, when the kingdom of God should come, he answered them and said, The kingdom of God cometh not with observation: Neither shall they say, Lo here! or, lo there! for, behold, the kingdom of God is within you.

Here He is admonishing the Pharisees not to sit passively and wait for the Kingdom to come, but rather to attend to themselves in order to prepare to receive it. Cyril of Alexandria (4th/5th c.) comments on this passage:

For ask not, He says, about the times in which the season of the kingdom of heaven shall again arise and come, but rather be in earnest, that ye may be found worthy of it, for it is within you, that is, it depends on your own wills, and is in your power whether or not you receive it. For every man who has attained to justification by means of faith in Christ, and is adorned by all virtues, is counted worth of the kingdom of heaven.

Commentary Upon the Gospel of St. Luke, Part II, Sermon CXVII (translated from the Syriac by R. Payne Smith, Oxford University Press, 1859), pp. 541-542.

The Greek phrase translated as "is taken by violence" in the King James Version is βιαζεται (biazetai), a passive form of the verb βιάζω (biazo). In the Greek, Matthew 11:12 and Luke 16:16 use the same word, βιάζω, conjugated differently. It conveys the meaning of achieving something by force. Both the KJV and RSV (the versions coincidentally recommended within the Greek Orthodox Church) use the word "violence" in their translations, but other versions choose something more mild. The the NIV, NASB, and ESV omit translating the phrase βιαζεται altogether, even though it appears in all available Greek manuscripts. Although the image of the Kingdom of Heaven being "taken by violence" may seem somewhat incongruent, the Lord Himself said that He came not to bring peace, but a sword (Matthew 10:34). The Russian Orthodox monk and writer Theophan the Recluse provided the following exegesis of Matthew 11:12, which, although written for Russian faithful living in the late 19th century could perhaps also be applied today:

The Kingdom suffereth violence - that is, it is attained with violence, with labor, force, and difficult spiritual struggles. Therefore, only those who lead a labor-filled ascetic life attain it. Thus, every sort of comfort is renounced along the path to the Kingdom. Pleasures of all types distance us from the Kingdom. But these days we have concern only for pleasures - sometimes emotional, but more often fleshly: to eat, drink, have fun, make merry, and luxuriate in everything. We have said to the Kingdom, "I beg you to excuse me," although there is a feast in the Kingdom - a royal feast - one so sumptuous that we could not even conceive of it, because we do not have the taste for it. What is considered sweet there is bitter to us, what is pleasant there is repulsive to us, what gladdens one there is a burden to us. We have gone totally separate ways. And the Kingdom, together with the violent who take it by force, withdraws from us. We are glad about this, and are even ready to drive them away more quickly. Indeed, we have already started talking about it, but the evil one has not yet managed to arrange this*.

Thoughts for Every Day, p. 133

* Theophan, writing in 1881, may have been referring to the revolutionary movements - all intensely secular - that were starting to take hold in Russia at the time. The editors note, "He identified the spiritual sickness of the modern West, with its symptoms of materialism, naturalism, deism, atheism, neopaganism, hedonism, and other antichristian belief systems."