I'll just tell you what I do with my 5 and half year old. I think create a routine, so she knows what to expect every single night. I put him in bed at the exact time every night. I read him a story and I stay next to him until he falls asleep. Then I have a clock in his room with colours. I tell him he can get up at 7 o clock not early at 3am or 5am. I also do a sticker reward chart. Think about as adults, we do our jobs properly we get a bonus. If kids are good we can reward them. I have the sticker chart where he gets a sticker for helping clean up etc and at the end of the sticker chart he can have a small toy. If he acts up I do tell him I will remove the sticker. This seems to be working for me.
Does she have enlarged tonsils? If they haven't already get the gp to check/ refer her to ENT. It's common for children with throat problems to wake in the night and I'm sure i read there's a link between adhd traits and enlarged tonsils/adenoids because they're not able to concentrate properly in the day with the lack of sleep. My friends daughter had hers removed and still has issues and now sleeps with a cpap mask at night and she's improved loads. She's also 5 &1/2 and never slept properly until this got sorted this year and she's like a different kid x