Actual live-in nurses are pretty rare and very expensive, but I think you’re overestimating how common they are.
Most of what you see is just caretakers who come during the week on a set schedule, which is usually a lot more affordable than nursing homes because you’re paying for someone’s labor versus labor + living accomodations.
To answer your second question, it would cost whatever rate the nurse agreed to work for. It would have to be pretty competitive. In most cases, having to live at the patient’s house isn’t seen as a benefit of the job.