Yes, I believe the agency should be informed, too. I find that it isn't uncommon for things like this to happen, where the nurse and client bond and cut the agency out of the loop except for the paycheck they collect from them.
Does the patient have an RN Case Manager? That is the first person I would have a conversation with.
Aside from that, even if nobody thought that unethical, the earth would have to be scorched and that duplex room the only building left standing before I would put myself in the position of being on call 24/7. I don't see them sitting there without a nurse for a call-off while your colleague is up there eating bon-bons and watching Lifetime Movies.
Maybe there is some sort of barter arrangement, which very likely amounts to the nurse having a second employer. If she wants to be a friend to the family, she should not be working there as a nurse.