I'd say give it a shot.
Go ahead and express to Jane that you are attracted to her and that you would like to get to know her better, exchange e-mail addresses (ain't technology grand?) and send her home.
You are right to be concerned about a long distance "relationship" and I would not encourage you to attempt to date over such a difference but there is no reason that you can't be friends.
Use the distance to your advantage.
At Our Age We Get Romantic Too Fast
Most relationships among your age group, which by the way is my age group too so I know what I'm talking about, get too romantic too fast. If you have long term (i.e. marriage) potential with this girl, you will need to be friends first anyway. Let that part of your relationship develop online.
Warning: Online communication opens up the possibility of presenting yourself as someone you're not. Agree in advance to be completely honest with each other about yourselves, your feelings and your opinions. Then anything that develops will be real.
Take it from a voice of experience, beginning a relationship this way can give it a real strength. One of two things will happen:
- you two will lose interest in each other, in which case your communication will naturally peter out; or
- your interest will be strengthened in which case you can worry about the next step.
For right now, rejoice in a new found friend.