I feel for you. It sounds like a lot of stress for you all, given the struggles of your relative, your relationship breakdown, MIL’s illness etc. Not to mention the usual stresses and strains of family life with two children. Is it possible you’re also perimenopausal and this is making it all more overwhelming and emotional?
There’s nothing more you can do. If you’ve explained to your SIL that you want to be on good terms, hopefully she’ll calm down and eventually be civil again. She was likely blindsided by your Ex’s attack and needs a little time to calm down and consider how she feels.
I’d strongly advise you to let the dust settle, with both your SIL and ex. You can’t push your SIL into being amicable again. Focus on yourself and your children. Start planning some things to occupy your time and look forward to: days out with the kids, meeting up with your own friends, fitness, things you enjoy (gigs, exhibitions, new books etc).
Have you sought any advice about moving forward without your ex? Are you even 100% sure you don’t want to try again? It must be very stressful for you all living in the same place. Do your children know you’ve split up? Do you have separate bedrooms or still sharing a bed together? It sounds like you’re still acting like a married couple, which is understandable, given there’s been no physical separation. He sounds like he’s under a lot of stress too (especially if his mum’s ill) and wanting to save your relationship.
If your relationship’s definitely over, for your own sanity, I advise you to start planning a future life for yourself. Even if it might be a while before you can afford your own home, start to make a plan to move towards that. For example, do you need to start looking for a better paid job?
Even if there’s a chance you might be able to patch things up, do more to build your own life and interests, so you can escape your shared home and claustrophobic family dramas more. Forget about his family and what they think of you for now. From odd things you say, it sounds like your self-esteem might be a bit low and you’re feeling too responsible for everyone. Focus on yourself, being a mum to your kids and what’s best for you.