I think the big thing for me is the time in the relationship she did it.
A lot of people are saying "if she truly loved you, she wouldn't have cheated". Considering it was a few months after you decided to start dating, and you had already mentioned about getting your head turned, I doubt either of you were "in love" with each other, there probably wasn't those strong feelings back then.
Now it's not like you've been dating a few months, you've been dating over a year, and you probably do love each other now, so now she realizes she cheated on someone she loved, either if she didn't at the time, and that's why she's admitted it to you.
You have to remember, it's a long distance relationship. I recently got married, to somebody I met online, who lived an 8 hour flight away from me. We talked, we liked each other for ages, but we never committed to a relationship, because they do fail a lot, because without that contact of seeing and touching a person, it can also make you feel distant emotionally, not just physically, and we didn't want to commit to each other until we knew that they were the only person we wanted to be with.
With your letter, stating that your head got turned, it may have been her thinking that you were interested in somebody else, that you needed that physical presence that you can't provide. With you guys not being "in love", she could very easily think it was just another online relationship soon to fail, like a lot of them, probably most, do.
As I said, I don't think it was her going behind her loved ones back, it was her being interested in you, and making a stupid decision because she thought that you wanted somebody else, somebody that was there, somebody that didn't have a child. Now she loves you, and feels so bad that she admitted it to you, because she now has those strong feelings for you. The only question I think you should ask yourself is "Can I get past this? Will this be a big problem for the rest of our relationship, and can she accept that I will have a fear about her doing it again". If you can accept all that, then stick with it, if not, then end it, because otherwise it will tear you guys apart eventually.
Also...