I think you have to accept there will be some tears and 'I hate myself' reactions, that you just have to let her express and then say: I know it's a sensitive subject but we have to tackle it. We need to work on you not hating yourself or feeling guilty or doing extreme diets because none of these are helping. We need to work on what would help.
One thing that really might help is finding exercise that she loves. especially one that burns loads of calories quickly like bootcamps, bodyweight training, weightlifting etc. Why not take up a fitness regime with her, and do it together four times a week, with a walk or swim or cycle on the other days.
Focus on lots of treats that are non food based - maybe face packs, bath oils, hair products, nail polish, hair accessories, magazines. Have one every time she completes a work out. Keep a progress report too, so she can see how much fitter and stronger she is getting. Focus entirely on this - fitness and strength - muscle tone, power, speed maybe rather than weight loss. Buy workout gear, take her for a new hair cut etc, so she gets lots of treats that are focused on a healthier way of life and an appreciation of her own body - looking and feeling good.
With food, focus on two things: optimum health - so you create dishes and snacks that are really nutritious, and as a less important things - any treats she loves that happen to be healthy or at least not fattening - eg fresh raspberries or strawberries, flavoured but unsweetened soft drinks like san pellegrinos. Encourage her to treat herself to things that don't make her binge.
Maybe suggest she keeps a food diary to see which unhealthy foods she eats most often trigger a binge or really aggressive cravings. see if she can swap those for treats that don't trigger binges. Tell her not to feel guilty if she buys and eats these treats. It's an experiment to see which ones harm her natural appetite most. this puts her in some control over the foods that control her, by having her study their effect on her.
It could help to do the opposite too. Work out which recipes and snacks help her feel full and lower the food noise. Make sure she has access to these at times when she feels like she might be more likely to binge.
Maybe also encourage her to practise saying things that will help, so that the phrases just come out instinctively in social situations e.g. if offered sweets or cakes at school, to say, 'No I am on a health kick' or 'No, sugar upsets my stomach'. If people pester her with comments like 'just one' or 'it's my birthday' or 'you just did a massive workout' encourage her to smile and say 'Thanks but no thanks. I really don't want one.'