I was poking around the Internet to see if there are any LDS-sanctioned modern translations of the Book of Mormon in English, and it seems there are not. (Although there's an unofficial one, A Plain English Reference to the Book of Mormon by Timothy B. Wilson.) However from another question on here I saw that the book of Mormon is still being translated into foreign languages. My question is, how is that the case? It was my understanding that Mormons say Joseph Smith returned the golden plates to the angel, so how are new translations made? From the English? And if foreign translations can be made from the English, why not a NIV-style English translation from the English?
You're right, the golden plates were returned to the angel Moroni at the conclusion of the original translation into English.
First, we have to understand the translation process (this link is the source to the quotes below).
There are some original translation manuscripts still in existence, but these are hardly legible anymore and, for several reasons (such as completing the whole translation in 60 days), contain many grammatical and spelling errors. Subsequent revisions of the English text referred back to the original manuscripts and other historical documents to fix technical issues and improve formatting. These have been published as new editions (most recently in 2013).
Regarding the relationship between textual revisions and the correctness of the translation:
With an English translation that is close the original writ, translations into other languages are based on the current English revision. Several other languages, such as Spanish, have been revised several times for the same reasons. Some translations are even incomplete and are only available as selected excerpts until a full translation can be made. (I know someone who is helping to re-translate the Book of Mormon into Persian. She told me that the current Persian edition has trouble with some of the word choice and sentence structure compared to the English reading of it.)
As you might imagine, these translations are time consuming and difficult to do well. The translation process is done by study, prayer, and faith.
Finally, to answer your last question ("Why not a NIV-style English translation of the Book of Mormon?"),
Well, Jacob Glad's answer added the missing piece, an official statement on the matter. Anyway, my original thoughts were:
If you want my guess, though, it's because the LDS Church wants to maintain the purity of the translation as much as possible. Translating into a different style of English may cause confusion, since it's quite likely elements of the text would inadvertently lose or change the meaning. As it stands, the language found in the current edition of the Book of Mormon is not nearly as archaic as the language in the KJV Bible, and is (in my opinion) still much more readable. With that perspective, some scholars suggest that an official, NIV-style edition of the Book of Mormon is simply not necessary.
There is a specific reason that the Book of Mormon is not translated into more modern English.
From a 1993 statement from the First Presidency of the Church,