Hi Sandra,
My xnh left me for another woman when my daughter was 15 (nearly 4 years ago). I understand the level of your pain for the infidelity (fortunately for me, it was not a friend or even acquaintance), and know that it would have killed me to have my d leave too.
I agree with what the others have said. You are doing the right thing to let her go--and try very hard to avoid the guilt trips or anger (easier said than done, I know). She is at that rotten age of 13 (much worse for girls than boys) and is trying to gain her independence. Her father has presented the perfect opportunity and she is jumping at the chance. She doesn't like your rules of behavior which her father does not currently employ so he can be "Disneyland Dad." My guess is that as soon as he has her around 24/7, he will either decide that he, too, must implement some rules, or send her back to mom. The girlfriend probably won't dig having her around all the time either as it will cut into her time.
I am sure your d will eventually realize that she misses her mom despite the rules and expectations of proper behavior. Whatever you do, do not stoop to your xh's level and indulge her when you're with her. She needs for you to be consistent in your treatment and expectations of her. Thankfully, you still have your son with you. Hold onto that for now, and I'm sure your d will eventually come back as well.
Many blessings,
Brigid