First question is what does your mum want? While there is any question of her going into res care there is this thing called "disregard" which basically gives your mum a bit of wait and see time to see if she gets well enough to go home. During this time, ss should pick up the bill for her res care. If she does go into res care, then yes her house will need to be sold to fund it, unless there is a dependent still living there, in which case, social service put a charge against the house and reclaim it when the house is sold. My experience is that generally, they understand that such things are difficult and take time and aren't going to be banging on the door with a big bill in a hurry. It may be possible for her to be at home with a big package of support, again it's all a bit wAit and see after a stroke
Hope this is helpful, shout if you have any more questions