It sounds like what she was really looking for was someone to rent her house while she was gone for a month and, incidentally, take care of her cat. If you were living there, I can see requesting utilities -- since you may use significantly more than she does -- but I would just adjust the rate I'm paying you and let you know why. Come to an agreement on what's reasonable, y'know?
I have a friend who has three cats and she was trying to do the same thing. She put in ads for people to rent the house for a three-month period while she would be out of town on a job. The deal was that the cost for renting the house for the three months was X amount of money, plus utilities, with the understanding that it was significantly less than other local vacation rentals because there was also pet care involved. I think she figured that she'd get a couple who wanted to stay there (she lived near the beach and a major city) and she'd get both some rent and free pet care. Didn't get many takers.
She changed the ad to read "Pet Sitter" and the info stated that the pet sitter could choose to be "live-in". She got a lot more bites on the ad, but never found anyone willing to pay part of her mortgage for the priviledge of watching her three cats. Finally ended up just paying someone to check up on them a few times a week like a normal pet-sitter.
Personally, I'd rather have someone like you, who's there to take care of the cat first and are just incidentally staying in the house, than to find someone wanting to rent out the house and then incidentally take care of the cat!