I would ask her following the points you have given.
Actually, before asking her to the ball, I think you should first lead in with "I've seen you around school, would you like to have a coffee with me after classes? Get to know each other?"
Asking her to the ball right up front might be too big.
Then, you two would be able to talk and if you'd still like to ask her, you're not complete strangers to each other.
Do you have a date yet for the military ball? I'd like to take you.
If she already has a date for it, then say "Oh, I guess I missed out. Well, maybe sometime we could go to a movie or something."
Then wait to judge her interest.
BTW, I don't understand how this girl is "really comfortable" around you, but you barely see each other. What do you mean by that?