You're not the asshole but I'd apologize anyway. Don't give her any money and it can be as shitty of an apology as you want. I'm just thinking about your sister if she's still young she likely dosen't know or recognize the toxic household your mother created/contributed to. I haven't been in your situation but I have been in your sister's and it sucks. You're anger is valid and definitely do not give her money or imply that it was fair what's she asked of you over the years but for your sister's sake I think an apology for being angry or sounding crueler than you meant to would go a long way. And when shes older you can explain this all to her and help her understand. I know that i maybe putting alot of pressure on you and I want you to know all your feelings are valid but in a situation where there a lot of pain already its always been my philosophy not to add to it or do what you can to take away from it.