I have three stepsisters who are, for a lack of a better word, just as selfish as your sister sounds. My choice is to cut them out of my life, and be curt and polite when I do deal with them. My older sister went to Mexico to do a get away wedding and only invited her new in laws. Her children have never called me Aunty or Aunt because I'm not related to them by blood. They show no respect to me, and as for such I am calm and chose to ignore them unless they show interest in me.

The best thing to do is distance yourself. You don't need to take that negativity from anyone, not even a family member. You don't have to offer any reasoning why. If she asks, say you simply aren't talking to her because you're tired from being sick. Done.