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To Asking For It: Please consider advising the father to have this kid evaluated by a pscyhiatrist. There are so many descriptions in this letter that remind me of my oldest son, who has Asperger's syndrome.
Often, kids who have this disorder seem "normal" to people (like the doting nanny) who've known them their whole lives and are used to giving them the line-item instruction regarding hygiene, cleaning up, etc. that they need. But these kids can be a nightmare for other adults, who know there is something wrong but sometimes can't quite put their finger on it.
Aspies kids have socialising difficulties and doing stuff like reading all the time is very common. They find it very hard to make social inferences from body language and facial expressions and they can seem really weird to someone who's not used to their flakier mannerisms. It can be really, really hard to bond with them because they don't seem to respond emotionally, and they often leave people with the impression they want to be left alone.
It doesn't help that Mom abandoned this kid, or she's been bounced around and spoiled, but I doubt that the kinds of hygiene issues described in this letter are because someone's waited on this kid too much. Fortunately, Asperger's kids usually do get better as they mature, especially if they have help recognizing and dealing with their shortcomings. They are often quite brilliant even if they are absent-minded as hell and could care less about stuff like their appearance.
Would the letter writer still feel like this was still a desperate situation if the kid was diagnosed with a problem like this and she was given specific advice on how to help? It's true that even raising normal teenagers is hellishly hard and sometimes unrewarding work, and Cary is right to sternly point out that parenting isn't a party you can choose to leave whenever things get boring or out of hand. But it might make the situation change dramatically to find out what specific steps need to be taken to help this girl, if she is diagnosed.
I think there is ample evidence that some questions need to be asked. Good luck and we wish the best for all of you.