I'm not sure about your last statement. the KJV probably played a big part in creating Modern English (not in "fixing"), but that also includes Shakespeare and yes, the OED later.
however, English is not "fixed" -- or else it would face the same problem as French which tried to keep the language 'contained' and ended up killing itself. English is continuously evolving, which is good for its longevity, but it's already seriously splitting into English languageS (just as Latin split into regional dialects that only by the 10th century really became separate languages
How it changed a Nation? hmmm... well, King James was Scottish by birth and nationality, and his reign was before the Act of Union in 1707, so it didn't change the United Kingdom as a nation. Did it have an influence on the Scottish and Welsh? Probably, but more so on the English. It did lay the groundwork, but the changing of the "nation" was really done by Henry VII's time -- in the 14th century, the English were still listed as "one of the Germanic nations" in church councils and at the time of James I, the population was still only 4 to 6 million in England and Walese with maybe 2 million in Scotland -- a very tiny country compared to France (25% of Europe's population) or Italy (Germany was still underpopulated and still heavily forested -- in fact prior to the late Middle Ages Germany was still heavily wooded like we see in "Gladiator"), anyway, I digress
So, the changing was done earlier, the KJV helped, and Oliver Cromwell (though I personally dislike the man) did the majority of the part in this change of national perception as "English". This was only reinforced in the later wars with France and raised to its heights in the Victorian Era.
Shakespeare's English is rather different from the English of today, yet it is much closer in time to Jonathan Swift's English than Swift is to ours. Swift is completely understandable to any English reader today. No footnotes are necessary for children being introduced to his work.
Dictionaries intervened.
ML/NJ