Ime the first prerequisite for being able to help a struggling teen is to have her trust. This doesn't mean you have to do what she wants or be "soft", but it does mean you have to be able to rein in your frustrations and avoid a punitive approach. This girl has a problem, it is quite clearly genuine, it doesn't matter whose fault it is- all that matters is how the two of you can get the better of it. You need to be on the same side. Kneejerk reactions are the last thing you want.
First of all, I would look at your general communication. Not because I am for a moment suggesting that there is anything wrong with your relationship which has led to this problem, but because you are going to need every ounce of trust and companionable-ness that you can lay hold of. Is there anything you enjoy talking about or doing together that has nothing to do with eating or exercise? Work on that. Make it clear that there are other things about her that you enjoy and value and which are still more important. Don't let her think she is the eating problem.
Secondly, I would not at all rule out the possibility that she is depressed. Extreme vivacity can be a sign of anxiety and/or depression. It takes my dd that way. Also, the slumping in bed. So I would keep in mind that there is at least the possibility that she has a MH disorder. Don't waste time speculating whether it's "real" or not; you will only exhaust yourself that way. The more you can get away from thoughts of blame the better: you will achieve more if you can think of this as a problem that the two of you have to deal with, like a sprained ankle.
Thirdly, do not punish her but tell her that she needs to see a doctor to rule out any physiological problem, like thyroid disorders. Don't let her argue about that, explain that some things are treatable and she could have more energy and that you will not take no for an answer. Stay as calm as you can.
If it does turn out she has an eating disorder/is over-eating due to depression, think of a two-pronged approach. Try to find a way of exercising that she would actually enjoy (drama classes, dance, some sport she might fancy, swimming). Ime very few teens enjoy country walks, especially with their parents. But also try to encourage her to get professional support. Make it clear that no one will be judging her. Make it clear that any treatment will put her in charge of her body.
But avoid kneejerk reactions at all cost.