This family member has told me they only need my help for 2-3 weeks. I've seen how being gone for that length of time can affect my teaching evaluations.
Could you limit your leave to one week and arrange for private nursing for the remaining time period? You could then make another visit to the family member at the next school vacation or at the end of the semester, to give them support during the later part of the recuperation. (Some local support options must exist as many people do not have relatives who can care for them during an illness. Although ideally one would always be able to support a family member in need, for whatever amount of time they needed, it is not always possible.)