your sisters illness has probably brought it home how fragile life can be and coupled with your boys leaving,that in itself is life changing as your role changes then too.so the ground under you starts turning to sand and you feel the foundations have gone before they actually have.
Is it just you your brother isn't speaking too in which case would it be an idea to write to him if he won't speak and suggest that whatever your differences are your mum and sister need both of you right now and they don't need to be worrying about you fighting so can you put it aside and show a united front to get through the coming months .If he doesn't respond at least you will have tried.
Why not talk to your husband see what he thinks about the idea maybe he would be able to talk too him more easily than you can.Other than that just take each day as it comes but don't neglect your own health,going to see your Gp might be a good idea,could be something your body is short of right now is creating an underlying depression separate from the events going on.