I feel for you, we've had a lot of problems with 3 year old dd and sleep...
Things that have helped:
Deciding on a routine and pushing it every day.
You could try offering an option of a nap before midday...but if she does not take it try to keep her awake until you want her to sleep.
Having exactly the same routine of calming activities before bedtime each night, eg bath, milk, bedtime story.
Initially setting bedtime to a point at which dd naturally fell asleep very quickly and then gradually bringing it earlier. The idea was that she learnt to associate the calming activities with going to sleep quickly.
Being quite mean about keeping DD awake when she is supposed to be ?e.g not letting her lie down and tickling her.
Once it is bedtime, I just keep telling her to go to bed?.. I try to bore her into submission.
Could you get anyone to help if you want to try this? .e.g with keeping dd awake when she is supposed to be.
With a routine she might wake less at night?or she might just be wanting your company ..or it might be a phase.
Things that didn't help:
Letting dd watch cartoons at 2am.
Letting dd nap randomly and late in the day - the problem was that I was so tired that I just let her do it...but this stopped me establishing a routine.
Good luck.