I wonder what your daughter's daytimes are like.
Is she getting plenty of physical activity during the day? (This usually helps kids relax better at night – toddlers have plenty of energy that needs to be expended.)
Is she exposed to plenty of full daylight, especially in the morning hours? (This helps set the day/night cycle in the brain, and allows the natural sleep hormone melatonin to be produced by the brain at night. Also, time spent outdoors in nature is known to have a calming, centering effect on children's nervous systems.)
Does she watch much television? (The passivity of TV, the fast editing, and even the business and intensity of the sound tend to disturb normal brain patterns, especially in young children who haven't yet developed mature brains. And the light is from the blue end of the spectrum, which can cut down melatonin production if the exposure happens within a couple of hours of bedtime.)
Is she exposed to environmental chemicals? (So much of what is found in modern household cleaning products, including scents, are actually highly toxic, and can have effects on the nervous system. If you do no other single thing to deal with this, at least stop using fabric softener on her clothing and bedding. That one change could make a difference.)
Does she eat much processed, commercially prepared food? (Several food additives, including artificial colors and a common preservative, have been shown in a large, well-controlled British study to cause more hyper behavior in children. It makes sense that sleep could be affected, too.)
Speaking as a person who has never been a good sleeper, your daughter may not be ABLE to sleep, and that isn't something she can control. What she CAN learn to control, eventually, is getting up, wandering around, and being noisy during the night. I'd give her options for the middle of the night: a small lamp and some books, a few cuddly "quiet" toys, music that she can play, etc.
That will leave you less infuriated, but it will probably leave her more lonely. So I'd work on this from as many other angles as possible. You might even ask your pediatrician whether taking melatonin at night would be advisable. Some kids with neurological issues benefit from it, but I don't know what age or doses are considered safe.