J.B.
For the child who has problems and must have constant care -the mother should contact social services. If she has no money, they will set up nursing care for the child -in home- while the mother works. If she has some sort of permanent disorder, the child can get disability. I'm not sure how all of that works, but plenty of people do it. Why can't the mother get unemployment? It sounds like she's not thoroughly investigating the opportunities available to her for aid. When children are part of the picture, it's easier to qualify for food stamps, section 8 housing subsidies, etc.
Unless she's educated with a background in some sort of technology that offers a lot of telecommuting jobs or education where she could find work with an online curriculum company, she's probably not going to fare very well trying to work from home. She also has to take into account the fact that if the child needs constant care (and what about the other two) -how much work can she get done while at home? I work from home two days a week, and if one (much less both) of my kids are here for some reason -it's very dicey as to how much I can get done. Plus, if the child has a chronic health problem that won't ever go away, some day she's going to have to find care for her. Those situations are very sad, but that's the reality.