Power everything down - computer and monitor both. Restart. Then go into the power settings and uncheck all the sleep stuff. Let the monitor run all the time the computer is on. If you have a modern monitor, there should be no "burn-in" problem. And as far as energy savings go, fuggedaboud that eco-crap. Insignificant.
Now, if it doesn't wake up after you power everything down completely (not just "restart") and wait 30 seconds to power up again... I have no clue.