Oh dear! I see both sides of this story very clearly. My early life (up to 14) or so was plagued by my parents' firm conviction that all children needed 10 to 12 hours sleep every night. I didn't. But as my parents packed me of to bed at 7.30 and got cross if I got out of bed again, I soon learnt to stay in my bed, or move very quietly indeed if I got out of it. Until I was 7 I had a night light and an electric heater that gave off light so I could read, lying on my tummy in front of said heater, with one ear sharply cocked to hear the sitting-room door open. The signal for getting back into bed pronto.
As an adult I have had to see things from my parents' point of view, although I have never been convinced that children do need 10 or 12 hours sleep every night.
If a child is just being naughty then rules must be made and adhered to, if she is scared of the monsters under the bed, or is having nightmares then these have to be dealt with.
You ask why she is not tired? Either because she is not using her brain enough during the day, or because she is not getting enough exercise. What does she do at the after school club?
How intelligent is she? An intelligent six year old is probably bored at school as she understood the teacher the first time something was explained, and is sitting there waiting for the others to catch up.
If you and the child's mother are convinced this is not just naughtiness, some of it is, after all, why on earth are you putting up with name-calling? That sort of thing should, in my opinion, earn her an even earlier bedtime for a week or so. I think her mum should speak to the school and club and ask what is going on there, and how the child is behaving there. She may be being bullied, after all and is desperately trying to assert herself at home in the only way she knows.