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)
Enneagram Type, Tritype, Instincts and MBTI Survey 2014
Enneagram Explorations is conducting a new study correlating Enneagram Type, Tritype, Instincts and MBTI. A link to the survey can be found at this SURVEY LINK and further explanation of the survey objectives can be found HERE. The code for the survey is Study 2014. The survey is open until October 29th.