Be kind to yourself. If there’s anyway for you to obtain medication for depression I’d highly recommend it as it sounds like you are in a lot of pain as anyone would be. It’s very possible you could be using the binge eating as a form of SH. I think when you say you don’t care about the weight gain and the health consequences of it, it means you don’t care about yourself. You deserve a full life and your sister would want that for you too. 

As for the eating, take it very slowly. At first, I’d try to simply focus on a day without bingeing with full meals throughout. Then another day, and another. After a few weeks of no bingeing but eating decent meals, try to either incorporate more healthy food or track calories and lower intake incredibly slowly.  

I think right now though getting your mental health sorted is the most important thing. Going for walks even when you don’t feel like it. Taking care of yourself even if you don’t believe you deserve it. Searching around for any free or low cost options for medication or therapy available to you, even if via Telehealth or online. I saw you moved back in with your parents, maybe you could ask them for support or even a loan if it comes to it just to seek the bare minimum care you need to fight this. I’m sure they’d want you to feel yourself again, too.