I would say her singing to herself IS a good coping skill. One of my earliest memories is being sent to bed what felt like hours before I was actually able to sleep. I'd lay there in the dark, trying to find ways to amuse myself, and afraid to call for mommy if I felt thirsty or sick or had to pee, because she'd be mad at me. But I couldn't will myself to sleep. In fact, the harder one "tries" to sleep, the more elusive it tends to become.
It may be a biological clock thing set in the brain. I'm still a night person, getting most energetic between 8 p.m. and maybe 1 a.m., and if I go to bed early, I often still lay there for an hour or longer. I doubt that your daughter can help it. But next time you take her to the doc, ask about using melatonin. This natural brain hormone can help bring on sleep (I use it when I MUST get to sleep earlier), and some pediatricians think it's safe for children in tiny doses.
It can also be helpful to make sure children get a good, long exposure to natural daylight every day, which can also help regulate the natural production of melatonin. And NO BLUE LIGHT (like from TV and computer screens) within at least 2 hours of bedtime, because it delays the production of natural melatonin. Lots of physical activity during the day also helps some children unwind and sleep better at night.
Behaviors are strategies to get some legitimate need met. In children, these behaviors are generally spontaneous more than intentional, and often their strategies are pretty confounding and ineffective, because they don't have the life experience that increases the range of choices. So, if your daughter were able to actually feel tired and go to sleep at night, she wouldn't have such long, lonely times in the dark to look forward to (or dread), and will probably not fight going to bed so much.
Also, if you can empathize deeply with her desire to stay up and play, she'll feel more understood and listened to. At which point, kids are usually able to cooperate better with what must be. If they're NOT listened to with loving care, they feel constantly manipulated, controlled, rushed, and thwarted, and many self-respecting kids will struggle against that.
Find out more about this dynamic in the practical little book How to Talk So Kids Will Listen, and Listen So Kids Will Talk, by Faber and Mazlish. The authors have led parent workshops for decades, and they offer tips and techniques that really work.