If I were in your place and thinking along those lines, here’s what I think I would do. I would stop by Jane’s office in a friendly way when her door is open and see if she’s free to talk for a minute—or just ask casually, when you see her getting coffee or getting off the elevator (but not in the rest room), if she could spare you a moment in the next day or two, and get yourself invited in.
Once settled in Jane’s guest chair (“Is it all right if I close the door?”), I would say something like this: “I keep hearing that one of the best things for a young person to do if she is interested in advancement in her profession is to find a mentor. Do you agree with that advice, and how would you advise me to do that? I don’t know how to go about it.”
If she is actually a potential match for you, she won’t turn this question away. (If she does, forget it, she’s not going to be much of a guide for you.) And now she is already mentoring you before even signing up for it!
Consider asking her about her own experience: “Did you find that a mentor was able to help you along early in your career?”
So then, at some point in the conversation she is almost certain to ask you if you have someone in mind. You know what to say then, right? And if she doesn’t ask you this (how could she help it?), you can lead the way with “How would you suggest that I approach the person I have in mind?”—and then approach her right then and there with what she has just told you to do.
Before you leave, ask Jane what’s the best way to let your manager know of your new relationship. Assume that your manager will know you’re in Jane’s office almost as soon as you sit down. Your manager is pretty likely to feel threatened by this, thinking you are going over his or her head with something. Jane will know what to do about this, so let her handle it.
Jane may also realize that if she is mentoring you, it will affect your relationship with your present manager on an ongoing basis. Let Jane solve that problem; make sure she knows you are aware of the delicacy of that, and ask for her help with it.