I think Joseph put the Book of Mormon into the religious language he was familiar with - that of the King James Bible. Any translator has the choice of vernacular to put a work into and he thought this sounded appropriate for a sacred text. Would like to point out that while the King James Bible has some inaccuracies, it is still probably the best English translation out there for preserving the poetic feel of the Bible. Second best would be the New Jerusalem Bible. The Bible, New and Old Testament, is not a prose document. It is poetry, 100%. If you really want to hear it, go to a conservative Jewish Synagogue some Saturday and listen to the chanting. In the original, there are sound-poetic elements that we are all familiar with in English - rhyme, meter, alliteration, assonance etc. In addition, the Bible uses many kinds of structure poetry couplets, acrostics and chiasm. (side note - Book of Mormon also has similar poetic elements, especially chiastic poetry) To translate one work of poetry into a different language is a new artistic work. Due to cultural factors at the time of King James Bible, poetry was considered as the only appropriate vernacular for sacred subjects.

Mormons today share that view of a certain vernacular being appropriate for sacred subjects and are taught to pray in King James English.

I think the answer is probably a lot simpler than some might think. Joseph Smith was basically illiterate, as his wife testifies to, when he set out to translate the Book of Mormon. It is possible he learned English from the Holy Bible itself, and used the examples therein as a basic format to go by. Considering many families at the time were poor and often the Holy Bible was the only book they owned as books were not easily come by.

LDS Perspective

If this is a truth question, one can only speculate because as far as I can tell, it does not appear that Joseph Smith ever addressed the question of why the Book of Mormon uses the English style it does.

Ultimately, the LDS perspective is that God commanded and helped Joseph to translate the plates so that translation was not decided by Joseph Smith himself, but by God. See Book of Mormon Translation - lds.org

Premise

The assumption that the Book of Mormon is written with the English of the Jacobean era stands to be proven. Comparing it to the KJV of the Bible may not necessarily be considered Jacobean English because scholars have said that 90% of it was based on the Tyndale translation which was done prior to the Jacobean period. Coggan 1968, pp. 18-19.

English in the Book of Mormon

Many of the stylistic examples in the Book of Mormon that may be incorrectly attributed to the style of English may actually be attributed to the original language of the text. For example,

Possession in English is shown in two constructs-"the man's house" and "the house of the man"-but only the latter construct is employed in Hebrew. The lack of apostrophe possession in the Book of Mormon is consistene with a translation from the Gebrew construct. Further, the "of" construct is common for adjectival relationships in Hebrew. Correspondingly, the Book of Mormon consistently employs phrases such as "plates of brass" (1 Ne. 3:12) instead of "brass plates" and "walls of stone" (Alma 48:8) rather than "stone walls."

Another example:

Sentence structures and clause-combining mechanisms in Hebrew differ from those in English. Long strings of subordinate clauses and verbal expressions, such as those in Helaman 1:16-17 and Mosiah 2:20-21 and 7:21-22, are acceptable in Hebrew, though unorthodox and discouraged in English: "Ye all are witnesses…that Zeniff, who was made king,…he being over-zealous,…therefore being deceived by…king Laman, who having entered into a treaty,…and having yielded up [various cities],…and the land round about-and all this he did, for the sole purpose of abringing this people…into bondage" (Mosiah 7:21-22).

Many more examples are given in this Study done on Book of Mormon Language

A quick text analysis of the Book of Mormon shows that the phrase "and it came to pass" occurs in the Book of Mormon 1123 times. One LDS member makes the case that the phrase "and it came to pass" is precisely what certain words in the original language means (therefore not necessarily an english phrase from an earlier era). See "And it came to pass..."

From an exmormon standpoint, an alternative explanation is that there were no plates to "translate," and the whole thing was something he created, using contemporary sources. The current thinking is that he (obviously) drew heavily from the King James Bible, but also several other books that were popular at the time and widely available in his area of the country. The books that appear to be likely sources are:

1) View of the Hebrews, a work by Ethan Smith which has a very similar "plot" as the Book of Mormon- that Native Americans are descendants of Jews that traveled from Israel.

2) The Late War, a book written about the War of 1812 for school children in New York. It was written in "biblical" prose, and many of the passages have striking similarities to passages in the BoM.

3) The First Book of Napoleon, another work that was contemporary to Joseph Smith, which also contains many passages that read almost verbatim to passages in the BoM.

A lot of fascinating research has been done about word patterns in the BoM and similarities between the BoM and the works listed above. (I have links to multiple other sources but I can't post more than 2 due to being new to the site, apparently.)

Edit: Ok, now I have enough street cred to add more links:

View of the Hebrews (just wikipedia)

Side by side quotations from The Late War and the BoM

Article about word analysis and The First book of Napoleon. Youtube link to a presentation within the article.