Sounds to me like your daughter is binge-eating because she's unhappy. Eating has become a coping mechanism.
I had similar eating habits as a child. I never gained much weight, as I vomited the food back up. But I empathize with your DD.
I am actually inclined to agree with the PP who said focus LESS on her weight. I know that seems hard right now because her weight is making her miserable, but I suspect her overeating is a symptom of her unhappiness, and you need to treat the cause.
I recommend counselling for her if she can afford. NOT - and I can't stress this enough - NOT counselling for her weight. If she brings up her weight as one of the things which is making her unhappy with her counsellor (and she probably will) that's her choice.
Overeating is a very mindless action. I also encourage:
-Mindfulness. She's not too young to learn to meditate and it can be an incredible tool.
-Exercise. I know that you have been trying REALLY hard to encourage her to exercise. But maybe a different angle could help. During my obsessive eating days I never stuck to an exercise regime because I was fixated on my weight. If I didn't see results instantly I became disheartened and gave up. I only learned to stick with exercise as an adult when I realised exercise is about how I FEEL, not how I look.
It sounds like you guys already do a lot of walking and cycling which is great. It needs to be part of a routine, a daily regular thing. It's a hard habit to form (but one of the best and one which will benefit both your daughters for the rest of their lives.) I know you're a busy working mum and don't get in until 5.30. Maybe a daily evening walk would be a good start? DON'T MAKE IT ABOUT YOUR DD'S WEIGHT. I suggest telling your DDs, plural, that YOU want to start getting in 5,000 steps a day (or whatever you think is a reasonable amount for a walk with your DDs) and YOU want THEIR help and support in doing so. Enlist your DDs as your supporters, but also get them excited about the steps goal. That something you can all achieve.
That way you can give your DD a solid goal to work towards - a certain amount of steps - that's something she has achieved at the end of each walk.
Exercise goals like that are much better motivation than weight loss, and they give a sense of accomplishment on a daily basis.
-Still as an adult if I feel myself slipping into the unthinking mindset which leads to overeating, I do something which grounds me in the present moment, in my body. Meditation and exercise are the obvious ones but I also find having a candlelit bath, listening to my favourite music, helps. Or doing arts and crafts. Help your DD find some mindful coping mechanisms to replace the maladaptive coping mechanism of mindless overeating.
I'm not saying don't bring her to a dietician. But please don't do it without also addressing the underlying issues which are causing her to overeat. I know you're trying hard & you sound like a lovely mum. But sometimes that's just not enough on its own - she's at a difficult age and clearly has some struggles.
My last point: you don't mention your own exercise regime expect to mention when you joined stuff for your DD. A great piece of advice I heard once for modelling good eating/exercise habits to kids is, we can't say 'Do as I say, not as I do.' They need to see us do what we recommend. We need to normalise healthy eating/regular exercise by doing it in front of them. Again, I don't know what you do, and if you find it difficult to fit a lot of exercise in I'm certainly not judging. But it could be helpful for your DD for example to watch you do couchto5k or the 30 day shred - not do it just to encourage her and give up when she does, but actually do it yourself. When she sees you manage it, it will seem more like a normal achievable thing normal people do - and talk about it in terms of fitness goals, like 'I could only run for a minute at the beginning, now I can run for half an hour' or 'now I can do fifty sit ups' or whatever. Keeping the doc on fitness targets above weight loss is a much better way to incentive exercise and the weight loss with come later.
Okay! That was a really long post. Just a personal issue for me. Good luck OP.