B.S.
I have to say that I agree with Sandy, she made some very good points about the child's health and how that could possibly be an issue if you're watching him.
I personally think $100/wk is a good price for a HEALTHY child. My kids are 2 and 3 and I'm looking for daycare for them right now and the cheapest in-home I saw was $85/child per week but most average $110-$130 or so. And of course daycare centers are much more, $150 and up depending on the services they offer the kids and hours there.
Look, I think your attitude is awesome in wanting to help this family out. It is unfortunate that they've got so many expenses and his health is so poor. That being said, don't sell yourself short on the care you are providing the child and the service you're offering the family. If he was able to go to normal daycare it would probably be more expensive because of the extra attention he'd need. That doesn't mean I'm telling you to charge a higher rate because you can, but don't be scared to charge a fair rate that is equivalent to other providers. You will be doing a lot of work and deserve to be compensated fairly for it.
Good luck :)