some children are just naturally early risers, my youngest who is now 7 usually gets up at about 5.45am and has done since she was small, all i can suggest is to make sure that she has lots of things to keep her occupied, a jigsaw that has all the pieces jumbled up on the floor ( we got to mixing 7 or 8 eventually), books etc.. and ignore her unless whatever she is doing is dangerous.
harriet finally learnt that she can play quietly and not to leave her room until we came to her, although now she tends to wake her sisters - working on that now lol!