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Some posters are correct that the girl is likely doomed if the LW leaves, but that does not mean she should stay.
This is a sad story because it's hard to imagine a good outcome for Anna. DeBlock306 pointed out that if the LW leaves her boyfriend, he will most likely continue to neglect Anna and possibly invite another girlfriend into their lives. I would go a step further and predict that he will resent Anna for ruining his relationship with the LW. These are two adults who seem to think everything would be just perfect if they weren't stuck with a screwed up 16 year old. If the LW leaves because of Anna, her father will blame Anna.
That said, the LW must leave. Those who are suggesting she step up and become the mother figure need to re-read the letter: She doesn't want or even like Anna. She resents her, blames her, criticizes her, and seems shockingly unsympathetic toward what sounds like the childhood from hell. The LW has also contributed to the neglect of this girl by allowing and (explicitly or implicitly) encouraging a situation in which her own father rarely even saw her. She says it "suited [her] fine."
If LW stays, she is not going to suddenly start encouraging the father to radically change his behavior so that Anna comes first. It's very clear that the LW would be very unhappy with a change in priorities. Even if she were so stung by our collective criticism that she vowed to try for a while, she would inevitably end up cracking from the strain of pretending.
My guess is that Anna is fully aware that the LW thinks she is ugly, stupid, and in the way. The LW's presence must be a constant source of stress and hurt for her. No wonder she spends so much time away from home.
I think we are all frustrated that the father did not write this letter, because we wish we could talk some sense into him. Anna is his responsibility, and he has failed horribly. But we can't write letters to the father. The only question before us, really, is whether LW should stay or go. She should go. If I were religious, I would pray for that poor girl.