Before I get asked, I'm her college professor and yes, there is an age gap between us that's pretty obvious. She has always been very flirtatious with me. The problem with her is that's she's very shy. I try talking to her and offering my help, but she gets extra nervous. She starts blushing, playing with her hair, looking at me directly in the eyes, and smiling uncontrollably. But is very difficult engaging in a conversation with her not only because she's shy but because she's super reserved with her interests. She's told me in the past that she took the class again just to see me. Now, I really like this woman but how can I show her my interest?
Your interest seems inappropriate until she is no longer a student, especially a student of yours. Perhaps give her a way of contacting you outside of class time such as your number, a suggestion of friending you on Facebook, or something similar.
Keep it friendly but appropriate to your professional role for now. Wait for the appropriate time to tell her you want to date her and not just be her professor.
Adding her on some form of social media after she's done with your classes so there's no conflict of interest. Not everyday a tutor/professor adds a student on Facebook, I would feel like something's up.
Ask her out for coffee in a public setting nothing too loud but be cautious as this could cause issues career wise. Go for it!