(post rescued from the soon-to-be-defunct Unconditional Parenting board)
A wise person once told me that on those occasions when you want to sleep and for some reason you can't (like you're on a night shift, or you're caring for a child who, for whatever reason, isn't ready to sleep, or is sleeping in a very unsettled manner) there's a curve of ghastliness, and if you can sit out the curve, about 15-20 minutes after you feel your head will explode, it actually becomes ok to stay awake. It might take a few similar experiences before you can recognise this.
Useful things: put on a DVD in a room with a mattress. You can lie and doze while child watches DVD. Don't expect child to go to sleep for 4 hours - if they happen to go to sleep earlier, then well and good, but it means you stop being impatient. Remember that while lying with your eyes shut is not as restful as sleep, it is much more restful than having eyes open and being alert. As Kipper says "One night without sleep won't matter".
In the time when you are awake, schedule yourself a nap for tomorrow, while child naps. Remind yourself that child will probably sleep in till 11am, and give yourself permission to do the same, cancelling engagements by email or text if you can.