It's good that you have your cats. They need your care and may bring some joy. You also have to get out of bed to feed your cats, that's a good thing. Maybe it's time to do first things first, and worry about losing weight later when you're not depressed.
Eating right is a good start. Also having a routine may help. Sometimes when you're depressed you feel tired and have 0 motivation to do anything else than sleep. Then you do nothing and your life starts to feel not worth living. I think it's important to break that circle, and force yourself to do something. It doesn't sound fun, I know. But please, try to think of things you normally enjoy.

Do you have a friend or someone you could go to the grocery store with? Eating is important. Try cooking or buy some healthy food and keep playing with your cats, they need you and love you.

I haven't done this myself, but my therapist keeps telling that writing down positive things or thoughts may help.