I'm going to go against the common wisdom here: since the girl in question has obviously had difficulty in previous relationships, I think her closeness / comfort around the OP are *good* signs. She'll be more likely to trust the OP than some random guy she just met, and given how much she obviously already trusts him, she'll be much more likely to see how the problems with past boyfriends don't apply here. I don't really think girls friendzone guys as much as we think they do -- plenty of guys friendzone themselves by never giving women reason to view them as more than friends.
OP, what you lack is confidence. Being sensitive and caring isn't the same thing as being passive and deferential -- you've got to believe that she likes you and show her that she should like you back, all while being sensitive to her feelings and boundaries. You don't have to come on strong -- but if, say, you're sharing a moment watching a movie on the couch, maybe scooting a little closer or even taking her hand would be about right -- and seeing how she reacts. Slowly but fearlessly -- if she reacts badly, just be a gentleman, calmly apologize, and say you'd misread her. But if she reacts well...well, don't freak out or push it too far -- just let things progress naturally while making sure she's comfortable with what's going on. And, for the love of God -- if a girl's looking at you like she wants you to kiss her, kiss her!