First, since this is your MIL and SIL, what does your DH want to do?
Second, if the two of you agree that some sort of support is reasonable and necessary, then here's what I'd suggest.
1. Figure out what MIL's actual budget would need to be to get by, how much she has coming in, and what the gap is. IMO, providing financial support entitles you to a full view of the financial situation.
2. Figure out the fair market value of the free babysitting SIL receives -- what would she need to pay to hire someone else to do what MIL does for free? Reduce this by some percentage to account for the fact that SIL is the local one and thus has to deal with all of the MIL issues -- 20%, 50%, whatever you think is fair given the number of ongoing problems (e.g., medical checkups) and personality conflicts.* SIL should be solely responsible for covering this amount of your MIL's needs, because she alone is receiving the benefit of the free babysitting (and handling the burden of dealing with MIL).
3. Subtract #2 from #1. This is the gap that needs to be filled by both you and SIL (because the subtraction of the babysitting value is just "payment for services received and rendered between MIL and SIL"). Divide this number by 2 to figure out the additional support that should be provided by each of you. If you and your DH agree that you can afford it, send that amount in assistance every month.
3.a Note that there is a decent chance that when you subtract 2 from 1, the result will be negative, i.e., MIL is providing more in free services than she needs to cover her bills. It sounds like there is not too much of an income gap to cover, and MIL is effectively providing on-demand, full-time babysitting, which is expensive, and so if MIL was getting paid for her services, she could more than cover her own bills. If that is the case, then you may reasonably decline the request. Either SIL can cover MIL's additional needs (in recognition that SIL is getting a steal of a deal on the babysitting), or MIL can offer her babysitting services on the market (where she would earn more than enough to support herself) and SIL can pay for her own babysitter.
*I love my MIL, but I'm pretty sure I'd need someone to pay me to actually live with her. ;-) Then again, beggars can't be choosers and all that.