Op, i can relate to what you are going through. But i have to echo what a few other people have said upthread - YOU can't do anything. The only person who can help is your sister.
I've known my best friend since we met on our first day of high school age 12. Well over 10 years. She's always been overweight. By the time she was 15, she was into the obese category and is now into the morbidly obese category. She's 5ft 3 and a size 28 (which is starting to look tight these days).
She's definitely not depressed though. She's 'loving life' as she puts it. She has zero confidence around men though and has never been in a relationship. She had sex last year with some random guy from a nightclub 'just to get it out the way' as she felt embarrassed about being a virgin at her age.
I can relate to the 'just loving food' thing. My friend loathes fruit and veg and this is why she has never attempted a healthy eating lifestyle. She says life is too short to be unhappy so she eats foods that she likes.
She works fulltime. Skips breakfast, has a hamburger for lunch from the local takeout van behind her work, and then orders a chinese takeaway each evening when she gets home. She also goes out drinking at the pub 4-5 nights a week, or will go to her friend's house for a 2nd dinner and a night long drinking session.
She doesn't do this because she's unhappy. She does this because it makes her happy. She loves eating, drinking, and socialising. She feels she would be miserable on a diet.
She has no interest in fashion and will happily live in hoodies and tracksuits outside of work. So being able to shop on the high street isn't a motivation for her.
She sits behind a counter all day for work so isn't getting exercise, and gets a taxi there and home each day.
The fact is, she just doesn't seem to want it enough to try. She bought some of those HerbaLife shakes last year but gave in after the first day because they were 'stinking'. Thank God, i thought (as i don't agree with them), and reminded her that a healthy bowl of cereal would likely have just as much cals as one of those shakes anyway.) Apart from when she brings it up - which is very rarely - we don't discuss her weight.
I'm worried sick she's going to die before we reach 40. Already she struggles to walk properly. She not even 25 yet. Her older brother and her parents are also obese. So i think my friend just doesn't know any other lifestyle. I spent a week staying at their house one summer when i was 14 and i put on half a stone!
But she is genuinely happy. Her weight doesn't seem to affect any part of her life.So she has no motivation to want to change. I think it will probably take a heart attack or similar before she wants to lose weight.
I'm so sorry for what you're going through. It's horrible seeing someone you love slowly killing themselves and not being able to help.
The difference between your sister and my friend though is that your sister sounds as though she is very unhappy being obese. So there is hope for her. I think getting her to focus on a job/hobby she loves will help to shift her attention from food so much.
I know that when i'm feeling bored, my first instinct is to reach for dd's sweetie tub. Now though, i write short stories for dd in the evenings when i have nothing to do.
Not quite the same as your sister, as i don't have depression, but i think it's important to focus on the positives - things that will lift her mood - rather than the negatives - things that bring her mood down.