To be fair to your sister (and I really don't want to be because she sounds like a PITA) it sounds like it would have been impossible to make your DM be sensible. I mean she was clearly so ill that non-medical strangers were wanting to send her to hospital and she refused.
I hope she's over the worst quickly, but it might be a long recovery. I had pneumonia when I was in my 40s and it took me weeks to get back to normal so at your Mum's age it might be months.
I am an absolute cow so I generally lean in to the type of person that your sister is (criticising other people while actually making things worse herself). I would say "Fantastic, it will be a great help to have you here, there's so much to be sorted out and Mum will need a lot of extra care. What date will I tell Mum that you are arriving?" (She's not coming, you know that, I know that and I don't even know her.)