Firstly, you are at the start of the period where DCs will want to stay awake until 9.30 - 10pm. It's absolutely pointless fighting it.
Up to bed by 8.30 and read, so adults get some evening and DCs don't start watching unsuitable TV, but you can't force them to go to sleep.
9 year-olds are starting to what some freedom and control over their lives, bed time today (as it's the obvious first step with light evenings), but there will be lots of others.
9 year-olds, know in their hearts they can't be left for long on their own, cycle to town on busy roads, stay out at their mates until 10pm, but a bit of them wants those freedoms of 11/12 now. The result is ridiculously small choices become important to them.
If you can give your DD a bit of freedom about bedtime, clothes, what or where to eat or any other daft little thing it helps enormously.
More than any other age preteens seem to be the ones to pick your battles.
A toddler forgets and my 15 year old understands adults have good reasons to be stubborn and sometimes make mistakes.
9 year-olds remember, bear grudges and still get in toddler grade tizzies they can't control. They are the most frustrating adorable mix, grown up one minute and little children the next.
Also the better they behave at school, the more this mix of emotions of wanting control and cuddles comes out at home.