Hello long time since my DCs were little but can remember the 'joy' of bedtimes around that age especially if DH was working away , (poor Daddy abroad in his hotel with room service and no kiddies, grrr!).
Anyway just a couple of suggestions: Start by looking at daytime activities and diet. Does your 3 year old do plenty of energetic stuff? Does she get fresh air? At this age she is probably like the duracell bunny - just keeps going.
Is she having a nap or quiet time during the day? Tempting as it was to let my DD snooze on so I could rest get on with chores, I knew if she napped too long I'd pay for it at bedtime.
Does she have chocolate or coca cola or hot chocolate? Even just a bit of caffeine can affect children. There's so much sugar in what we eat it all adds up and it's worth keeping an eye on the meals and snacks she gets.
When does she start to slow down, get calmer? I wonder if by the time you put her to bed she's on her second wind, so starts up again? Any chance of getting her to bed earlier?
I know people always say bath and book before bed. I sometimes found it better to vary that a bit - it doesn't matter what order you do things as long as it's quiet time. So in summer if DD came in hot and sticky she'd get her bath or shower as soon as possible, tea after. Then a quiet activity followed by toothbrush-cuddle-bed. Boring, quiet.
Dim lights, stop tv sooner than usual, try and get things quiet ahead of bedtime. There's always the sticker + reward ploy if you think that's worth a go?
You've probably tried a lot of this but it really is repetiton and not rising to the bait of paying more attention to wilder antics. Sorry for the essay.
Please help I have a 5 yr old and a 34 month old and they share a room. The youngest just thinks bed time is a joke and becomes incredibly hyper jumping around and laughing and it ends up bring an awful experience. If I try to be firm with her she gets worse and thinks its hysterical. This is all after a bath and one story each. My husband works nights so I am always doing this on my own. I feel completely useless as I've been trying to sort out bedtimes since they were born. I can never get anything else done in the evenings as by the time they eventually go to sleep I have often had to fall asleep with them too and it's so late I am exhausted. I've read the books that say bath story and bed but have never been able to get it to work with mine.