Personally I would be massively pissed off about this and I don't know if I would be able to deal with it. There are not 2 of you now, there are 3. You, DH and his Ex. She will be calling the shots of getting maintenance and when he can see her. You won't be able to plan Christmas, birthdays or holidays now without putting her needs first to make sure she is fully included.
@AutumnFabreeze the mother's apparently not interested in money, and I doubt very much she wants to lose her Christmas and other special days with her child, either. It sounds far more as if the child wants to get to know her father a little, that's all. You're assuming an awful lot about what this 12 year old wants, and it's diametrically opposite to everything that we know about the mother, to assume that she wants that level of involvement with her child. And the girl is 12 already - the days when parents have to mediate contact are decidedly limited!
How about the OP takes it one step at a time, and sees how things pan out?
OP - you may like her. She may be a nice girl, and she may adore your sons, which is surely a swift route to your own heart! She may of course be a little madam and a menace, I'm not going to be sentimental about it, but... if she reminds you of your own kids, which if she's got the same father she may, then you may find it easier than you think to like her. And she may be anxious for you to do so, too. Just give her, and the situation, a chance. Let it settle. Don't jump to dire conclusions all at once.
I do sympathise, honestly. I'd be weirded out as well if an ex of my husband's suddenly got in touch with this news, and I'd also feel a bit threatened. But it is what it is, and for all you know, your sons have just acquired an older sister who will love them for life. It might not be bad. Just give the girl a chance.