1st, so very sorry about your DGC and hope they are well soon.
As for your sister, well, I have an idea why she is estranged from her AC and having trouble in her marriage. I hope those situations improve, but it doesn't sound likely.
She may have been advised to fill in the void her AC's absence has left by socializing with others, focusing on other family, etc. But that hardly means she has the right to demand attention or complain about not being included in your holidays, etc. It's not your job to make up for her lost or failing relationships. I understand her reaching out, but she needs to accept whatever is feasible.
So I agree with Monica - let her know when you're available and stick to it. She may argue, push for more, and play the victim card - in fact, she probably will. But if you stick to your guns, over time, she'll have to accept it.