Oof.

To begin, I'd ask if LW and mom are from a culture where children are supposed to personally look after elderly parents, and where sending mom to a care center would be seen as LW's moral failing. Perhaps some of mom's resistance to care aides stems from there?

I'd also ask if sister can and wants to care of mom. If sister has no family then she might be better suited to care for her, as long as sister gets proper support to avoid burning out too.

In some of these "children must look after the parent" type of cultures, the child who cares for the parent also receives a bigger part of the inheritance to compensate them for the time and effort spent. If that's the case here, is getting a bigger chunk of the inheritance the reason why LW hasn't involved her sister? Because she could have asked her sister to take care of mom for a while, or to contribute in any other way.

I'd also love to know what the heck LW's husband contributes to the household, because as far as we know LW is the one taking care of everybody else.