Locally, we pay a teenaged/young adult sitter $10/hr for a healthy child. I would pay your friend at least that much, possibly $15 or more, especially if she is dealing with such a messy illness.
I would also talk to the daycare because if the diarrhea is caused by the medicine and the child will have some level of problem for the duration of the medication, is there ANY provision to send her? In a pull up, perhaps? I would not send a child if acute, but my DD can have a good day and then maybe loose bowls the next day til her meds are done. Def. give your kid bananas and yogurt, but I'd still talk to the director if this is totally about the meds and not that she has a nasty stomach bug that is contagious.