J.G.
If he goes to sleep at 11 and you let him sleep til he wakes up.. when does he wake up:?
Use that as your gauge....
some kids just don't take naps anymore at a certain point.. I had one quit by the time she was two.
he is old enough to understand needing quiet time though. Tell him he must sit quietly on is bed, look at books or do a quiet activity for 1/2 an hour. tell him he doesn't have to sleep but must be quiet.. the "reverse psychology might actually encourage a nap!
So let's pretend he wakes up at 8 after going to bed at 10. He needs 10 hours of sleep.
Start by establishing a routine. Same thing in the same order every night. Bath (try lavender baby bath and lotion), warm milk and a bowl of oatmeal, quiet story time, white noise in his room (fan/humidifier), warm his blankets and pajamas in the dryer. Tuck him into bed at 7:30. If he gets up - gently put him back to bed. If he cries - leave the room and set the time for 5 minutes. Go back after 5 and tuck him back in. Set the timer for 10. After 10 minutes if he is still up. tuck him back in and set it for 15. Do it again for 20 if he has not fallen asleep yet. If he is still awake.. change is diaper or make him go to the bathroom, give him a small drink, rub his back for a bit. Then do that all over again. The idea is he can cry himself to sleep (it won't hurt him) or learn that you will come back and fall asleep waiting. You can do this for 3 or 4 nights... so be prepared. He will eventually get it.. and you have to be firm. Each night it should take less time until he has learned the routine.
He must may not need naps.. and need less sleep. If you need the break.. consider saving TV time if he will sit through a show.. for a time when you need it. Or consider asking a neighborhood 11 or 12 year old if they would like to come over and play with him.. take him outside for fresh air.. or on a walk around the block or nearby park. Most young girls this age LOVE toddlers and many will do this for free.. it will give them experience for a reference for future babysitting jobs.. or if you can afford it hire someone to come for an hour each day. Consider enrolling him in a preschool to help burn some of that extra energy.... and give you a break too.
best wishes..