It sounds like a lot of damage has been done and you're going to need to change the way you handle her, to make any progress. Often the strict approach is not effective, especially for a difficult child, because we're asking them to comply at a level they just can't handle. It's like punishing a newborn baby for not telling us in words what they want.
A book we recommend here is "The Explosive Child" by Ross Greene. It's a different way of handling a child, it almost turns previous methods on their heads. You can use the techniques on easy child kids too (difficult child = Gift From God, the child that brought you here, easy child = Perfect Child, although none really are THAT perfect!).
I don't personally know any kids with Turners Syndrome, but I do know about it. And from what I know, I suspect that even the experts are still floundering, when it comes to what each kid is capable of and where the child can achieve (or has difficulty). So in this, you will probably have to work out a lot of it for yourself. Sorry. It really will dpeend on where she can perform, and where she just can't do it.
As far as discipline goes - try Explsoive Child, you really have nothing to lose. And the really good thing about it - if it works for you (and I think it has a good chance) then it will work for anyone who uses the same techniques and is on the same page. Anyone NOT on the same page, is suddenly going to seem like an ogre. It works by changing the way she sees you. You are no longer the strict disciplinarian (because it didn't work, did it?) and instead, you are the support, the facilitator, the assistant. She sees you in the role of helper and she learns to go to you for help and to work WITH you.
There are something you need to ignore - just do't react to the disrespect or shouting. But instead, show respect and don't shout, because YOU are the hero who is showing the standard of how to behave. You then have every right to quietly say, when she is shouting at you - "I'm not shouting at you; please don't shout at me."
Much of how it works is in the preliminary set-up. You try to put things in place when she's calm. Of course when a kid is raging, they often forget plans and strategies, but it's a matter of keeping your cool and helping her regain her composure before you try to get anything more out of her. Teaching her to communicate frustration in more effective, acceptable ways is perhaps a big target, but it makes a very big difference when you can get to that stage.
The aim of all of this - to get her to learn to behave more appropriately. And it seems from what you say, that a lot of her bad behaviour is likely to be due to frustration, anger (at past mismanagement, especially if her father was overly strict) and possibly difficulty in maintaining self-control at the moment.
There are going to be some areas where she just can't perform at the same level as others of her age. This is frustrating to her as well as you, and from what I understand of Turner's (I could be wrong, because my information may be faulty) there are going to be some areas where it will always be alien territory. What you may need to begin to focus on, especially as things get more challenging academically, is finding a career path for her and putting it in place now. Again, this is very much dependent on how she is going and what she is capable of. Another thing about genetic defects - there can be a very wide variation in how the condition is presented, simply because there can be anything form a partial loss/added chromosome, to full expression. A Downs girl we know is considerably handicapped, at 15 she still can't read. And yet another Downs boy we know, same age, has been doing really well in school. There really is a wide range in capability, every kid is different.
On the subject of her wanting to stay up late to do homework - if it IS homework she is doing, I would actually let her do it. But I would talk to her teacher about the time it is taking her. easy child was about 11 when she began burning the midnight oil doing homework and assignments, because she hadn't organised her time properly. I remember getting up once at 2 am because an assignment she had been working on, was lost when the computer suddenly crashed and she hadn't done a save for a few hours. I typed, while she dictated. it was the quickest way for me to get back to bed and get sleep! But easy child did learn, as I let her stay up, that if she did her homework late at night she would be tired andcranky the next day. She learnt fairly quickly, as a rsult of natural consequences, to use a bit more careful organisation.
Maybe your daughter's teacher could help her organise herself a bit better with homework? Help her set up a homework plan, for example, so she's not needing to be up so late at night. But this shouldn't be a battle for you, to this extent. By talking to her teacher about it, instead of you being part of your daughter's problem, you have become part of her solution. She is not as well equipped as you, to solve problems. She needs to know that you can be her problem-solver, or at least help her learn how to find her own solutions.
You will find h her doctors can help you with medical issues, with hormone levels and other stuff (all very important for her medical care), but are not going to be very useful when it comes to the behavioural issues you're facing. And even the usual psychologists - their yardsticks are people with a full complement of chromosomes, and frankly a lot of their measures are not really going to be as relevant. What you deal with, is what you deal with.
Be wary of labels which define her behaviour - they may not be a comfortable fit. She is her own person, she has her own qualities.
I wish I could be more help, but what we CAN do here, is provide support and ideas for ways to handle specific behaviour problems, regardless of the underlying cause. And that's why I love that book so much - the diagnosis doesn't matter. It just dives in and works on te behaviour, in ways that are ridiculously easy to implement, compared to the alternatives.
There are other books, also. They're all worth a read. And before you fork out money for any book, try and grab a copy form a local library. There is some discussion on Explsoive Child in the Early Childhood forum, it can give you a bit more idea.
Welcome to the site.
Marg