I think it really depends on where you live and the age of the babysitter. For a 12 year old, I would think $5 to $6 an hour would be fair. I would assume this would not include any late nights.
We usually have high school girls....15-18. We pay $8 to $10 an hour (for 3 kids...although we paid the same for 2...it is still time) (we live in Northern CA...in LA and SF areas I hear much more is expected). Usually ends up being about $10 with tip if house is picked up and dishes done. I really like to have the same babysitter, so I try to pay a bit more then they would expect, so they will come back. When I was in highschool, over 20 years ago, my neighbors paid me $6 to $7 an hour...which was a lot back then, but I was always available for them :)) To me that is important.
I also don't really take into account if the kids are awake or not (or how many kids). She is still at your house watching your kids and her time should be worth the same. Everyone has some down time at work and you aren't paid differently, you still have to be there.
Ask around your neighborhood and see what the going rate is. It really is different in different areas.