A.S.
What I have leaarned from my children is, they do not sleep like adults! With most children, the later they stay up and the more tired they are...the less they sleep. Try putting your daughter to bed earlier and see if that makes a difference. Also, you might try cutting out that AM nap. I know it's hard when she gets up at 5:30. I am sure she is exhausted by 8 am! However, if you get her to bed earlier she probably won't be so tired in the morning.
My boys go to bed at 7 -7:30. If they stay up later they will be up before 6 am. If you get them down before they get overly tired it will be less of a fight too. If she is tired at night, why try to keep her up? There have been nights when my boys are tired and they go down at 6:30 and sleep until 7.
These are just things I have discovered with my boys (14, 5 and 3). I also do daycare and see the same things with the daycare kids. The parents who keep their kids up at night come in saying their kids were up at 5 am.
Good luck to you!!