Apparently Dell LCD monitors, at least the E207WFP, but probably others, have a fairly delicate power button. Over time and use, the part of the power button that actually contacts a circuit board is pulverized. Once this occurs you can't turn it on (or off, depending on your luck).
This happened to my Father-in-Law's monitor. Being the good son-in-law, I swapped his with mine, and brought mine back home to do something about it.
Prevention: Turn on the monitor and never use the power button again. Just turn off your power strip.
Unplug the monitor from the computer and power.
Remove the front bezel. CAREFULLY, because the ribbon that connects the button circuit board is very short. You'll have to get something thin between the bezel and the case, and gently pry the case out to release the bezel. This is a pain and rather difficult.
With a tiny phillips (sp?) screwdriver, remove the very short screws that hold the circuit board to the bezel. Remove the smashed power button, leaving the other buttons alone.
Carefully put the circuit board back on, and carefully replace the four screws. Don't overtighten!
Snap the bezel back into place. It goes in pretty easy, making up, a little, for the difficulty in removing it.
Plug the monitor back in. There's a hole where the power button used to be. That square thing in the middle of the hole is the green power LED. Don't mess with that. To the left of that, barely visible through the hole, is a small metallic circle. That's the actual power button. Use something with a rounded point to push that. You'll feel it. The LED will come on, and the monitor will function just fine. The other buttons work too. I'll probably cover up that LED, it's distracting.
Look up at the Prevention step. That's the best way to work from now on.
Repair: If it's under warranty, call Dell. If not, see Workaround, above.