When your DD said that she didn't realise she had gained that much weight, in your shoes I would have said something along the lines of, 'well if you want to lose it, you know what you need to do, don't you? Depending on her response, I would then have perhaps gone on to say, 'I can help at home by making sure we only have healthy meals, and cutting down on the treats that we keep in the cupboard, but if you eat stuff you know you shouldn't when you're out with friends, then only you can do something about that'.
So maybe when she's stopped sulking, or however she's behaving because she's upset with you, you could have a chat with her, and say something along the lines of 'Tania, do you want to do something about your weight, as you seemed quite upset about it earlier?' Assuming she says yes, then go on to say something along the lines of what I suggested above. Don't be critical of her, she knows she's got a problem but get her talking if you can, and then encourage but don't nag.
Also, does she get much exercise OP? If not, do you know why? As you could also attack the problem by pointing out that if she were to do more exercise, it would not only help her lose weight, but would also make her more healthy. I've always had weight problems, but at one point in my life I found a video exercise program which I found easy to do, a lot of them I found were too fast, so by the time I'd got my head around the movements, they'd moved on. Anyway, this video took me only 20 minutes to do each morning, and not only did it help me lose weight, but it made me feel better within myself, which then lead to me quite often wanting to do it a second time, either later in the day, or straight after the first session. All the time that I was doing that, I could literally eat whatever I wanted, and didn't gain a pound.
So at the end of the day, you can encourage her to eat more healthily - although you cannot monitor what goes on when she's with her mates, and you can talk about and encourage her to find a form of exercise that she enjoys, but you CANNOT do it for her. Maybe you could suggest putting on some music and having a dance each day, or something like that, as a lot of youngster enjoy a good bop, and it's easy to fit in at any point, day or night.