Awww! Dealing with guilt and anger is so hard because even though you're down, your mom still needs care. It's tough!
Have you talked to your mom about how you're feeling? Maybe have a real heart-to-heart with her about the things that she feels she needs more of. For example, if she says she needs you to do this or that with her, maybe what she's really saying is that she wants more of your attention. And how lovely would it be if the two of you could spend time together doing something you like to do, rather than HAVE to do. Know what I mean?
As far as your sister...my biggest piece of advice is to tell her how you're feeling. ASK for help. In my experience, saying something like "You may not realize it, but it's been really hard on me being responsible for mom's care 24/7 and I need a break" in a non-accusatory "You should have known!" tone, works well. And be open to hearing her thoughts on how she can help. Maybe she can't give you tons of time, but can send money towards paying for respite care so you can get out. If she flat-out refuses to help, then be mad. But give her a chance to do right. She may surprise you.
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