I was married to a man who was abandoned at he age of four by his mother.
She was the child of an violently abusive, alcoholic father and a meek mother. She was not happy in her marriage to my husband's father, and ended up having an affair. When he found out, he dragged her around in front of the children by her hair, calling her a whore and an unfit mother. She moved out of the house, and he filed for divorce and full custody of the kids.
In her guilt, shame, and unwillingness to constantly fight with her soon-to-be ex, she thought the kids would be better off just staying with him, so she dropped (my husband) off after visitation one afternoon and never went back. They didn't meet again for 16 years.
I have seen how damaging this is from both sides. She was suicidally depressed because she couldn't deal with issues from her own abusive childhood, and felt like anything she did was too little, too late. He has definite issues with abandonment, fidelity, and anger. After his mom left, his father went on to marry an absolute bitch who treated him and his brother as inferior to her own children, and made it well known that any problems that she had with his dad were due to my husband's very existence. It was not his mother's fault, but he always waited for her to save him, and was never told anything about why she left.
It's been a hard road, but from what I've seen, you're doing the right thing now. What else are you supposed to do? You can't go back and undo these last few years!
If I were you, I would write an even longer letter, explaining your thought processes, past living situation, and precisely why you have been gone for so long.
Sure, she might be angry and not want to hear any more right now, but at least she'll know about your perspective on things and that it definitively was NOT her fault. Thirteen is really not that old. You still have time to get to know each other again, and build a new relationship.
In these comments I hear a lot of anger from single mothers and previously-abandoned children, but after having been one and loving the other, I can tell you that love and support is the best way to heal the gaping holes left by this situation. Obviously, it's been hard for your ex and daughter these last few years, but like I said, you can't go back. You shouldn't be expected to spend the rest of your life groveling because you were in a bad situation that you didn't know how to deal with. Really, I have SEEN what happens when a non-custodial parent constantly disappoints the child, and if I had to choose, I think I'd rather have the one who didn't constantly make empty promises. Besides, who in their right mind would deny their child a chance to have a healthy relationship with thier bio-parent? This is a golden opportunity for all of you to move on. As long as you're not a toxic person, you can be very beneficial to her emotional wellbeing.
All any of us can do in this life is try to learn from past mistakes and do better in the future. I think you should send your letter, and be very clear that you will wait forever for her to write or call when she's ready, and then carry on with your life. If she refuses to forgive you, there's nothing you can do, but you'll know you've done your best to try and fix this.