Eowyn -- at least as performed by Miranda Otto -- strikes me as an ENFP. It's been a while since I've read the thing, so Tolkien's version may differ. Mostly what I remember is her complete change of romantic allegiance, and indeed personality, within a paragraph.
Tolkien wasn't that great with characters. He was more a theme kind of guy.
I'd say Gandalf the Grey is INTP, and Gandalf the White leans more toward INTJ.
Sam: ESFJ or ISFJ
Legolas: ISFP, INFP or INFJ
Sauroman: INTJ or ENTJ
King Theodin: ESFJ
Bilbo: INFJ (on the outside he seems ISFJ, but he can plan like a crazy man the way only NJs can)