The fact that she confessed immediately makes me think that she truly is sorry, and, more than that, wants to stay with you. Especially in a long distance relationship, it's very easy to hide things.
Personally, I would put more emphasis on that, that she is so honest, even with tough situations, than what she did when she was drunk.
Judgment is the very first thing to go, when you drink.
I personally would be forgiving of her. This doesn't sound like a game she's playing, and she does seem completely sorry for her actions. What more can you ask for?