LadyRabbit - if you want to help her, the place to start is her mental health.
I was bullied at secondary school, and have a lifelong history of depression as a result - the bullying led to me having terribly low self esteem - to this day, I am surprised that people like me.
I have a voice in my head that has basically taken over from the bullies, and that is so harsh and critical - I call myself a failure, berate myself for every bad food decision, lie awake at night with the voice telling me how dreadful I am - why would anyone like someone as gross as me. When my therapist asked me to bring an image of myself to mind, the critical inner me came up with Jabba the Hut - that is how I see myself and how I judge myself.
Basically I do not like myself AT ALL - so why would I care enough about that person to feed them well, or encourage them to make tiny steps towards healthier decisions?
Maybe your sister feels like that. If she is filled with similar self-loathing, then you having a go at her will be horrendously painful, and might serve only to reinforce her belief that she is unworthy, worthless, a failure.
Until she has tackled her depression, she will struggle to deal with her weight. I am. It saying she has to be completely cured before she can start dealing with her weight, but finding some inner peace, and a kinder inner voice is the starting point.
I am having cognitive behavioural therapy, to treat my depression, and it is also helping me make some small steps toward healthier lifestyle choices. Basically, I am having to learn how to be kind to myself.
Remember, it is very easy to use food to smother the sadness inside - I know I do that. Carbs can have an almost sedative effect - I have used them to pack away the feelings of unhappiness, if that makes sense.
It is also worth remembering that our brains have not evolved to cope with the sheer level of abundance of food that surrounds us. How long do,you think we have had supermarkets offering thousands of different food products - many of them so cheap? I don't think we had anything near this level of abundance when I was growing up - it is very new. Our bodies have evolved over thousands of years, and for the vast majority of that time, most people have lived hand-to-mouth - if they were lucky, they had enough to eat. So our brains are conditioned by those thousands of years of evolution to eat, eat, eat, when food is available, because it might be gone tomorrow.
But now, it isn't gone tomorrow - there is still plenty- and our brains need to catch up. That is why it can be so hard to lose weight - we are fighting our basest instincts.
Add in the fact that so much very processed food is addictive, so our bodies crave it, and you have another level to the struggle.
Remember also, at her weight, she will find it hellishly difficult to exercise - I get back ache and sore joints if I try to go for a walk. When I joined a gym last year, I was doing 3x10 minutes on the treadmill, and some resistance work - and pretty soon, my joints hurt so much, I couldn't walk up to the gym from the car park - so I stopped.
Runesigil - I see why you are suggesting the OP should ask her sister to help design a fitness programme for her - but there is a big risk she will see that as very manipulative - I know I would have - and my reaction would have been, 'I am fat, not stupid' - in addition to the huge hurt that even my nearest and dearest were judging me. The OP will know if it is an approach that would work,with her sister, but it would have had a very negative effect on me.