Hi S.
I ama teacher and have had to deal with similar situations, though not exactly like this. I hope that the school where teh child attends knows about the current situation, but if not please encourage the mom to tell them. It is important for the teacher to know so he/she can be aware of behavior changes, a need for time alone, etc. There is hopefully a councelor or something similar in the building that can help if needed.
I think the best advice I could give would be to let the child guide you. When one of my students lost a toddler sibling to cancer she never said a word about him being sick or dying. She did write me a little note the next year telling me why she missed the end of school, but that was the first time she even mentioned a problem (I was aware of it though). Her sister, however, talked alot about it to me when she was in my class. They both delt with it in very different ways.
I think that some trips out may be a good idea. Maybe a "girls' trip" to the mall for lunch or something, to a movie or too a park. Keep trying if she turns you down though. Some kids feel guilty having fun when a loved one is sick or has died. They feel like they should only be sad. She will need time.
I am glad that you are taking the time to think about the child. It will help to take some pressure off the mom, too, who will also need some time alone. Many mothers/fathers in this situation have been so wrapped up in taking care of their spouse and then worriing about thier children that they neglect themselves and dont' allow themselves to grieve.
What a sad situation. Please post more if there is anything else that we can do.