This list is just a few compelling reasons I've found. By no means is it exhaustive.
(1) The KJV is part of the Christian historical heritage, and its language in woven into the writings of the past 400 years. To not know its language means that the reader will be unable to appreciate so much of both Christian and secular literature. The culture of the Occidental world is based on Christian thought and the KJV.
(2) Valuable reference works are keyed to it. Concordances, commentaries, sermons, etc (many of which are in the public domain and available to all Christians) use the KJV. Most of the most solid and classic works which have withstood the test of time use the KJV.
(3) Italics indicate words added by translators. One or two other translations use italics, but most don't. This is one idea which should never have been abandoned.
(4) The older pronouns and verb endings are more precise. The Greek language is fully inflected and has many verb endings for gender and number. The modern English language is not as precise as Greek. The early modern English of the KJV still kept many of the middle and early English inflections that have since been dropped, and therefore translates more precisely.
(5) The KJV is a superb literal translation. One example is the precise translation of the historical present tense. This tense occurs when the text begins using the present tense in a past account for immediacy and immersion. Most translations use the past tense, because in English this historical present is not used. (The NASB is the only translation I know which flags the historical present, although it is not translated.) The reader must know the verb endings to pick up on this.
(6) The KJV is proven to be doctrinally correct over the course of centuries of the most intense scrutiny. What other translation could be agreed upon by Calvinists, non-Protestant Baptists, and Pentecostals?
(7) New is not always better. C.S. Lewis called it "chronological snobbery" - preferring the new to the old just because the new is new. We've witnessed the rise and now downfall of "dynamic equivalence" translation, which interprets as well as translates. With the wide adoption of the ESV, and the continued use of the KJV (and NKJV), the market for dynamic equivalence Bibles is shrinking. But think of everyone who adopted dynamic equivalence Bibles over the past 30 years or so, only to find them inadequate.
Conclusion: If it wasn't for the language barrier, there would be no reason to stop using the KJV. It is tried, true, and well-studied. Reference works are keyed to it, and the most solid commentaries go along with its text. It's difficult to improve on something that has been so well received and is such a part of Occidental culture, and to which so much of Christianity is tied.