The first question will be the easiest to discuss: Why only the KJV? I'm pretty familiar with KJV-only-ism from Protestantism and I have a hard time seeing how any of their reasons would apply to Mormons.
Its a question that has engaged Mormons for at least a few generations. Reasons can vary between Mormons from the "only" camp to the "retire it" group. There is arguably nothing definite in Mormonism that requires the use of the KJV as a final Bible authority. Indeed, one of the criticisms raised against Mormons from some Evangelicals is that Mormons don't respect the Bible enough. This is because Mormons don't believe that Bible is infallible or inerrant, but still of religious and spiritual value. It is the Word of God, but written by fallible humans in mortal language. Joseph Smith actually preferred a German translation while trying to understand the Hebrew underlying the Old Testament. Its rather amazing that the KJV has come into such singular use.
For over 100 years the KJV has been the one Bible Mormons used as an authoritative religious text. The reason seems to be out of a shared tradition with the larger Christian Protestant America who only in the mid 20th Century started developing a wider selection of popular translations. Mormons could have followed easily enough, but Mormon First Presidency leader J. Rueban Clark wrote "Why the King James Version" whose arguments had great influence. Most of the arguments for staying with the KJV can probably be traced to his book. One of the main arguments was that with all the flaws found in the KJV, other translations reduced the Messianic Lordship of Jesus Christ as Son of God, or downplayed miracles and prophecy.
Still, the main reason it has endured is because of the language used for translation of the Book of Mormon, and revelations found in Doctrine and Covenants and Pearl of Great Price. Joseph Smith, regardless of how amateurish the language to trained observers, used the KJV almost slavishly in putting down the words of inspiration. It is what Mormons have come to expect from Scriptures. As explained by an article devoted to the subject of Mormon use of the KJV:
Jacobean English is the language of LDS revelation. Furthermore, Joseph’s use of the KJV acted as a trigger for many of the revelations in the D&C . . . Joseph was fond of placing new doctrinal wine in old KJV bottles and much would be lost if Mormons were to use newer translations . . .
By retaining the KJV, a biblical link with certain Mormon doctrines is maintained and allows “all scripture [to be] woven together as one book” (Joseph Fielding McConkie). Use of another Bible would orphan Mormon phraseology, from the “And it came to pass”-es and other Jacobeanisms of the Book of Mormon to the important doctrines . . . Thus, to maintain unity with Joseph Smith and the Restoration, the KJV is retained.
The final main reason that the KJV dominates as the official Bible translation in use by the Church of Jesus Christ of Latter-day Saints is the 1979 publication of an extensive cross reference and study edition. It takes the idea of shared modern Scripture links and makes it in a sense literal. The project won some praise such as the 1980 top graphic arts award for the typesetting. A standard was set that could not be reproduced by easy replacement.
Another possible reason that no other translation was used, and by that time there were some that still retained a connection with the KJV, is copyright issues. For years the LDS Church had used Cambridge University Press version and there was no reason for that to change without some complications. Similarly this happened when the Spanish language Reina-Valera Bible was published, as stated in a question and answer, "The 1909 edition was chosen because of the availability of the copyright and the widespread regard for the quality of its translation."
Even if Mormon theology and history contains a non-mainstream Christianity, it essentially is conservative. Changes can happen abruptly, but only after necessity demands innovation. In a religious context it took more than a thousand years for the Restoration of the Gospel and the revelations of Joseph Smith and his successors were not received until questions got asked and pondered. Reasons to remain with the KJV might be slowly disappearing, but there currently isn't any overpowering reasons to switch. Members are free to use whatever version they would like in personal study. Even some LDS General Authorities have quoted from modern translations without any need to apologize. Regardless, the KJV continues to be used for unity of scripture, doctrine, membership, and tradition.