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I think one thing is left out of Cary's advice: the nanny. Neither the LW nor the so-called Dad are able to raise a child. Anna's behavior is not atypical of an unhappy teen. She needs help, but I don't think we need to assume she's got Asperger's.
The girl needs two things: she needs to care enough about herself to take care of herself, body and mind, and she needs not to expect that she will be cared for as if she were a child. Both those needs can be filled by a loving, caring adult WHO DOES NOT THINK OF HERSELF AS THE CHILD's SERVANT!
The nanny, if she's really the only responsible person in the picture, needs to understand that she must teach Anna to do for herself, not simply do things for Anna. The nanny, as an employee, and one who probably feels sorry for the girl, may not see it this way, but perhaps could be convinced to see it that way. Perhaps on her way out the door the LW could have this conversation with Daddy and Nanny.