E.L.
Whoa, I'm of a completely different opinion than some on here. Your daughter needs MORE sleep. She needs to go to bed earlier -- like 6:30/7pm. Do not let her put herself to bed. Children are not disciplined to self-regulate that way. I know that my daughter (who is the same age) would also stay up if we let her have books and a light. When you leave the room, the lights should be out and she should be quiet.
I lay down with my daughter at naptime, because she's too squirmy to fall asleep on her own during the day. She is taking a 1-1.5 hour nap, and now I get a 20-30 minute nap too. :) Or at least I lay down for a while. I just lay there until she's asleep and then I leave. I do not lay down with her at night.
When she wakes up at night, tell her to go back to bed. After she starts going to sleep earlier you will see a reduction and disappearance of those night wakings.