This broke my heart on a couple of levels. First, I can relate. I wasn't diagnosed with autism until I was 48 and until then, I pretty much faked adulting. Granted, I was pretty good at faking it. I was a snigle mother to 3 boys and I felt that same sense of failure and inadequacy.
Second, it breaks my heart because I really think you're being too hard on yourself. You've had a lot to deal with, way more than I did as a single mother. I didn't have a chronic illness, so there was that.
You can only do what you can do. You're teaching your daughter self care by pacing yourself and not taking on more than you can. Jesus, you've jsut had surgery, or rather, more than one.
Your Sophie sounds like a great kid. Let her help more. Let her learn, it'll give her confidence and something to take pride in. As you get stronger, you can do more.
You're not a bad mom. You love your daughter and you're doing the best you can. I rasied mine in a trialer with very little and now at 32, 29 and 27, you know what they remember? Well, they don't remember the bills or how shitty I felt. They remember the love, they remember the times we made scrambled eggs or pizza. They remember Family night on Fridays when we'd watch Sabrina The Teenage Witch and eat treats and dance to the one CD I owned at the time (Brooks and Dunn). They remember curling up in bed and reading. You've got this.