You can put a preschooler down for a nap, but you can't make her sleep

Through three children I have become a pseudo-expert on sleep. For example, I know that not getting enough sleep can change one's personality. (Granted, you don't need to read books to learn that, just stay up late a few nights in a row.) There was a study with a small group of children who had been diagnosed with Attention Deficit Disorder. Before putting them on medication, their sleep was adjusted so that they were sleeping more and with consistent routines. After the sleep adustment only 1 of the 13 or so children still showed symptoms of ADD. I also know that if you get close to sleeping while driving or while in a boring meeting it means you are not getting enough sleep and your body is trying to catch up.

The Curly Girl just turned 4, so she is getting to an age where napping ends. However, she still clearly needs more sleep than she is getting. She falls asleep on twenty-minute car rides, and by 4 in the afternoon she is screechy, whiny, and alternatively clingy and aggressively pushing us away. I know in my bones that she needs more sleep. But she goes to sleep at the same time no matter when we put her to bed; she gets up between 6 and 6:30 no matter what time she has fallen asleep; I put her in her room for a "quiet time" in the afternoon hoping she will fall asleep but she rarely does.

I have the feeling that I should know what to do, but it's like when you are just barely forgetting a word that's on the tip of your tongue. The solution is just out of reach.