Your daughter sounds very much like my 15 year old son who has recently been diagnosed with autism. He had problems in school from about year 3 or 4, but they became a lot worse when he started at secondary school. He had no friends, was ostracised. I spoke to school several times, but they weren't very interested - because he largely behaves himself at school and is very bright, they couldn't care less that he was unhappy. I also raised our growing suspicion that he may be autistic and they dismissed me.
In the end, we moved him for year 10. His new school have been fantastic, they really seem to understand him and how his brain works. I think he'd been there three weeks when someone rang me and asked me if I'd ever considered that he may be autistic. They referred him and we had a first consultation with an ed psych who essentially agreed that he was highly likely to be autistic.
Then the problems in the new school started - he had misread some social cues and hugged a girl when he shouldn't have - her boyfriend assaulted him in retaliation and all manner of bullying and spreading horrible rumours started.
The school handled it really well, but at this point, we decided to get him assessed privately and he is confirmed as being autistic. The school have been really good - he has been withdrawn from some GCSE classes to give him time in the wellbeing room, where he can get some time out (he has been struggling with panic attacks and has missed a fair amount of school) and catch up on other lessons. Since halfway through the first half term (since September), he's been much more settled, going in every day and has just sat his GCSE maths and numeracy.
I think a diagnosis has definitely helped, but having a school who are onboard with it and willing to put in place whatever support he has needed (as well as dealing with the bullying really well) has been just as important - probably more so. He still doesn't have any close friends, but gets on well with some people and seems a lot happier.
Good luck and I hope things improve for your dd