Thank you those of you who are reminding me about second chances - of course this is true, and I will try to keep it in mind.
"If the behaviour is only bad in some lessons, is there an underlying cause perhaps? Either in terms of disliking those subjects"
She is worst in lessons she finds difficult - finds difficult because she hasn't done the work, not because she finds them intrinsically difficult. So that's maths primarily, science and languages. Subjects where her native wit can carry her through - English, drama, sociology - not a problem (although she still won't do homework). She seems to cope very badly with having to try at anything which involves any sort of prolonged or difficult tasks, or anything she can't do easily. She has a massive failure of gumption.
It has got me thinking in the past about ADD, but I presume that this would have been considered by the psychiatrist. Also wonder why it wouldn't have manifested itself in primary. She's spectacularly disorganised, but on the rare occasion she does any work her presentation is beautiful. In maths (when she can be arsed) her working is really excellent - logically and consistently set out in a way that I certainly never managed as a child. Surely this wouldn't be there in the work of a child with ADD?
"OP, you mentioned in response to one comment that you have a son who is autistic and can be disruptive. Has there been any suggestion that your daughter feels that he's getting more attention, and it is disruptive behaviour that gets him it?"
She is jealous of her brother because I am close to him. This is an issue for her, but just one of her issues. He's a good kid.
"Tell her to do whatever she wants but will get no financial support from you or your family and no support in terms of day to day living."
Legally we are required to support her and keep her in school until she is 18. School leaving age goes up to 17 in 2014, and 18 the year after (I think). So we can't disengage completely.
"The other is to consider HE, maybe using something like InterHigh"
When we first took her out of school the plan was to home school her for a while while we looked around for another school. She basically refused to do anything other than watch tv, play games online, or lie in bed (if I denied access to tv and internet) with a pillow over her head. I nearly had a breakdown as I had the local authority on my back asking to see evidence of what we were doing at home. Hence signing her up with a tutorial centre, where she at least engaged with the teachers. I would have been quite happy to have allowed her to do any subjects she wanted at home, create her own projects, follow her own interests. Except she hasn't got any interests other than Facebook, watching TV and fiddling with her hair and nails.
"Have you tried specialist Adolescent counseling OP?"
No. How would I access this?
"I think a solution would be to let her work in a menial job"
If I could let her leave home and work as a chambermaid for a year, believe me, I'd jump at the chance. It was what I did at 17 after dropping out of my A-levels, and it totally pressed the 'reset' button on my attitude to education..... But she is 14. Nobody would have her. And she wouldn't agree to go.
"If she were mine I would either come down on her like a tonne of bricks or completely disengage"
We've had 2 years of this behaviour now, and I have done both of these things, fruitlessly, at different times.... If we come down hard on her she ramps up the aggression and the defiance to a point where it becomes dangerous for everyone in the house. If we disengage she creates situations (like now) where it becomes impossible for us to maintain this disengagement. Like staying out late and not telling us where she is, walking out the house in the middle of the night after a row, playing loud music at midnight on a school night, refusing to go in to school. It's honestly impossible to deal with. You can't disengage when your child is regularly school refusing or putting themselves in danger.
"What do you actually say to her when she says the teachers are incompetent?"
I point out that good behaviour and co-operation is even more vital in lessons where the teacher is struggling, as the learning still has to be done! That she owes it to herself and the other children not to take advantage of weak teachers by creating conflict, and that it's cruel and self-defeating to target teachers who may be finding their work more than they can manage.