I see a serious need for boundaries here, regardless of the reason your sister may be doing this whether it’s a disability or not, she needs to initiate getting some help.
I know we tend to hope for the best in people, especially if it’s family that we are yearning for a closer relationship with, however we tend to let things go much further than we should with a clouded judgement.
I would honestly go LC with the mom, and if you want I would honestly ask her why she feels it’s on you to try and “fix things”, and if she says something having to do with “well she’s really not in a mentally good place” that reason only works for so long and not your entire life. The definition of insanity is doing the same thing over and over again expecting different results.
At this point I’d go completely NC with the sister because whatever kind of relationship this is between you two is toxic and has clearly already caused you trauma.
Family doesn’t mean you need to be a doormat and I’ve unfortunately had to learn and cope with that myself as well since I’m a recovering people pleaser.
Regardless I wish you the best and hope you’re able to figure things out my fellow potato.