Billie,
I really feel for you and your DD. I only skimread your other thread (in a rush) but I felt I couldn't not comment. When I was 15-16 I had a similar thing. Going to school became gradually more and more of a problem. I would, like your daughter, get really stressed to the point of being physically sick. For the worst four or five months of it, I threw up in the bushes by the side of the road every morning on the way to school. I liked my school, I was doing very well academically, and I wasn't being bullied. It was, for all intents and purposes, a phobia of going to school. (which, for me, also extended to a number of other social situations - I didn't see the inside of a cinema for a good 5 years, I'd just get stressed and panic if I tried to go see a film).
Looking back, I think the problem for me was pressure. I'm a perfectionist, which is both an asset and a great flaw in my character - I never feel like I'm doing well enough. My results were great and all my teachers were happy, as were my parents. Still didn't feel good enough. On top of that, the social pressures of being a teenager in a school - I can be socially awkward and I just felt lost trying to navigate it all. Combine that with a serious bout of depression, and a bad reaction to being on the pill for the first time (how was I to know my body can't cope with oestrogen-containing anything....), and you have one seriously messed up teenager.
It went gradually - much as I'd love to, I can't point you to one specific thing that will make it better for your daughter. What really did seem to help was little things that took the pressure off.
After a while, the fear of having a panic attack in public was almost as bad as the things that caused them in the first place. So, it was arranged with my teachers that if I started to feel panicky, I could just walk out the class and go hide in the loo (as often I'd need to throw up), no questions asked, no hassle given if it took me 20 minutes to come back. I had a very dear friend who I trusted enough to tell him what was going on, and when I felt panicky, he'd make a point of just keeping talking to me to distract me - about the lesson, films, last night's football match, anything. I came off the pill, which was the best idea I had that year as they were really messing with me.
One of the big things for me was a sense of being unable to escape - in an exam, you can't get up and walk out if you get stressed and feel sick; in class, a teacher might not let you and even if they do everone will look and wonder why you have to go to the loo again. Obviously I don't know your daughter, but it's possible she feels the same.
It's important to realise these thoughts aren't rational in any way. I couldn't help it, and I imagine your DD can't either. Also, it's only now, at a ten year distance from it all, I can analyse it like this. I couldn't have explained it at the time.
Good luck, to you and your DD. I wish I had a magic wand I could wave to make it go away, I've been there and it's awful :(