J.,
I do not know the appropriate salary to give her, However in my opinion $360 a month is too low! If you do the math, you are essentially paying her $19 a day 5 days a week?????
I understand that the room and board is included but I would look at that as a bonus of sorts for her having to be on call. I know money is tight for everyone but if you could, I would bump that salary up a bit more. You say you want her to become part of the family and thats great but your including her in family fun time isnt going to pay her bills...and on that note I do not think you should be taking into account what her private bills are when deciding how much to pay her. I think you should do some more research, maybe a call a nanny service and ask some questions....all I know is that I would want the person responsible for taking care of my child at any and all hours of the day to be happy not bitter.---I am sure there are websites out there that could give you some answers...have you googled it?
K.