If writing and sending the letter will give you closure, then that's fine. But it doesn't sound like it would. You say that you'll lose your pride if you send it and she doesn't respond, which is very very likely. You're still caught up in what she thinks of you - will she now see me as needy, will I look stalkerish, etc.
And if she does respond, what answer would you possibly accept?
If she writes back with an explanation like CatAmongst's post above (which I think is very good), you won't consider that closure, will you? You'll want to respond - well you might have seen my invitation to stay the night as clingy but actually I was genuinely meaning you could just crash here, as for that gift, I'm just a gift-giving sort of guy, it wasn't like an engagement ring, etc. There's absolutely no point in opening that dialogue.
If she writes back and says "sorry but you were crap in bed", you'll know for sure that she's not a nice person, but will it give you closure? Seriously, what answer could she give, except "I was wrong and haven't stopped thinking about you", that would help? It's a key question, I think.
To be brutally honest - she probably doesn't think of you much at all. Right now she's not thinking of you like she underestimated you, your dignified silence means maybe she was wrong after all...she was relieved that you didn't push it, and now she's moved on. A letter is probably going to make her think 'oh, for God's sake' and then ignore it. But as I say, I can't see what response she could give that would help. So I think, leave it.