Every teenage girl who is fat knows she is fat. Home is supposed to be a safe space. People aren't supposed to call you fat in your safe space. OP, you and your husband need to stop talking about her weight - you might not use the actual word 'fat', but that's the only word she'll hear when you raise your 'concerns'.
You say she's told your SIL she hates the way she looks - what more evidence do you need to know that she doesn't feel comfortable talking about her appearance with you? Take the hint.
If you really want to help your daughter, lose weight yourself. I don't mean that in a horrid way - more that if she sees you making changes to your diet and lifestyle and sees you losing weight, she'll see that it can be done. She has half your genes, so if you can do it, she can do it too, especially as she's younger. Don't nag her, instead just silently become her role model. Trust me, you'll achieve so much more with these actions than with your words.
If she sees you shifting the weight, she'll start thinking about whether she has the motivation to do the same. If you start cooking healthier foods and they taste nice, it won't be a hardship for her to have some of that instead of the junk. Get everyone to buy into it - you don't say if your husband is a healthy weight or not, but get him to eat healthily too rather than eat junk food around her. Don't make a big deal of it, and don't make all the changes overnight - be more subtle about it. Sneak healthiness into your household so it feels natural, rather than a pointed attack.
Don't judge her (and you can judge without words, no eye rolls), don't talk about it, but do make your overall household healthier so when she's ready to lose weight, the tools to do so are all around her. That means, she'll be able to start doing so quietly and easily, without a fanfare. Often we're scared to change, because we're scared to let people see us fail, so don't pass any comments. Let her attempt to change her lifestyle on her own terms, without worrying about your reaction.
You never know, if you bite your tongue for long enough, one day she might feel safe enough to talk to you about her health. Let her come to you.