I really don't know what to suggest. At the risk of sounding ignorant, I'm pretty smooth when it comes to social encounters, so my suggestion would be to just be 'natural'. If you're convinced that doesn't work (or if it's so entirely vague that you just don't want to sort it out), then be casual, as Rad said. Ask her out, vaguely. Don't make it a clear-cut weapon and be like "Hey, you're really cute, and I really want to ask you out."
Try to notice what it is she brings in with her to the library. Does she bring in coffee? Invite her to the local coffeehouse to discuss literature. Roll with it.
EDIT: Please tell me you're not in this for sex.
EDIT2: Flower with a note is risky. If the girl was a bit insecure, then this would probably be a somewhat decent move, but I'm not getting that kind of a vibe from you. Wait awhile before you make a move that big. Be like a lawyer: "Don't ask a question you don't know the answer to," unless you're willing to take the risk.
Try to notice what it is she brings in with her to the library. Does she bring in coffee? Invite her to the local coffeehouse to discuss literature. Roll with it.
EDIT: Please tell me you're not in this for sex.
EDIT2: Flower with a note is risky. If the girl was a bit insecure, then this would probably be a somewhat decent move, but I'm not getting that kind of a vibe from you. Wait awhile before you make a move that big. Be like a lawyer: "Don't ask a question you don't know the answer to," unless you're willing to take the risk.