Hi talicat, sorry you are having a hard time. You definitely aren't alone - there are lots of parents here struggling with teens who are struggling, trying to work out what to do for the best, and feeling powerless.
It sounds like your daughter might have a few different things going on - physical ones and emotional ones. You might find it hard to disentangle everything.
Physically - It is most likely to be hormones, given her age. But also, just in case, ask the GP to check her for anaemia and vitamin D deficiency. I am just recovering from those myself, and the symptoms were very similar (feeling sick, off food, lethargic, losing interest in things/feeling down - I had other symptoms too). It is quite common with teenage girls, esp if they have heavy periods and/or don't eat a balanced and varied diet. When the blood test results come back look at her Ferritin and vit D levels - the 'normal' range will be shown next to the result - and even if she is not 'officially' below the normal range (and so anaemic and/or vitamin D deficient), if she is low/towards the bottom end of the ranges, get her some iron and/or vit D supplements. I have been on them for nearly 2 months now and feel much better: frankly I can't believe how bad I felt for such a 'simple' reason and how easily-fixed it has been. So it's worth checking for your DD.
Emotionally - it can be very hard to get emotional or psychological help for teenagers. Partly that's because there is a shortage of services. Partly it's because (as you have already discovered) no-one will see her if she is unwilling - there is (rightly but sometimes frustratingly) no 'compulsory counselling'. However, you can keep trying. When you go back to your GP, ask for a referral for counselling. Ask for it for yourself too, because you probably need it, and it will help you support your DDs better as well. Your GP should refer you without any problem, but it might be harder to get a referral for your daughter. (Why do you want to see another GP today, BTW?) If you can't get an NHS/CAMHS referral, then you could consider getting her private counselling or looking for a children's charity that provides it (there are a few around - Google the name of your town plus 'children's counselling').
There's a lot more I could say about support for you - which is absolutely crucial if you are alone and struggling with a 'troubled teen' - but that's a whole other thread! :) Keep coming back here though, because even if you are very alone in 'real life', you won't be here, and you can get lots of support. :)