Nice job Jumpan, get the dude even more in trouble with the law.
How well do you know this girl king? I'm talking in terms of years known, average time spent together (i.e. 3 days a week, etc.) before your house arrest, etc.
In my experience, letters while nice and romantic in tradition and classic movies are not really appreciated in real life. From what I've seen/been told, women prefer more direct communication (i.e. face to face talking). I guess they are able to tell if you're being genuine or not.
I also agree with Opium that any kind of letter has to come from the heart, so while you can ask for general advice, I wouldn't use specific advice (i.e. exact verbage and the like).
But what do I know? I don't have a lot of experience in the relationship department...
I've been in similar type situations before (well never under house arrest but still) and I'm not good at talking to women myself. I'm always seen as the friend/brother type. I guess women want you to be more direct with your feelings/intentions soon after meeting you else they will default to thinking of you as a friend only (and I'm living proof that it is quite difficult, perhaps near impossible, to go from just friends to a relationship).
Just go with what you feel is right. You said she doesn't have a car and you obviously can't leave. Why not offer to have a cab/limo/etc. bring her to your house (if you get approval of whomever you're living with since you mentioned it wasn't your place)? That's assuming you can afford that anyways (I dunno if you can or not).
Best of luck man, women are a mystery. The main reason (IMO) why a lot of guys have crappy luck with women (myself included) is that the women themselves don't even know what they want yet they expect us to know. Go figure eh?