I would meet with the mom and discuss what her immediate needs are. As much as I understand it's a rough situation and I know you feel horrible for the family, you still have to run your business.
I would start posting for drop in... moms wanting to socialize etc. Or a temporary child. Sometimes they're out there.
I would also take a good hard look at my budget and decide what I need to make things work. Then make my decision on the family from there.
But make sure the family knows that you are concerned about their child, but they need to be aware of your business. You'll help them in any way that you can help them. But there are always limits.
I hope the little one gets better soon.