I am not really giving an answer here, but rather a clue that may help in finding the answer.
I looked at the LXX (that includes English translation) for Psalms 22:21 and have noticed that, it too, uses the term "unicorn".
The Greek word that is used in the LXX (Ps 2221:21) is monokeros (μονόκερως). Using a Greek-English dictionary I learned that monokeros translates to "unicorn".
Regarding "Was this a mis-translation in the original King James version? Or does "unicorn" seem to be a valid translation for this word?" - It appears as though the KJV translated from the LXX and, if so, then the translation would be valid. I think the question begs whether or not the LXX has used monokeros properly in its translation.
Just throwing it out there.