So, we had a short power bump the other night and now I cannot get my AirPort Extreme to come back on. I have completely unplugged it and tried the reset button with no luck. This happened months ago as well, the difference was that after three days of being unplugged- when I plugged it back in it did come on and eventually reset. I'm at a loss and very frustrated- help!
As there is no power switch or external fuse on the airport line of hardware, the absence of any lights when you plug it in means that the hardware is failed.
If you have AppleCare on the computer it might cover that airport so definitely check that if you have a covered computer. If not, the repair cost for an airport is generally not much less than a refurbished or new router. Since there's not much that even a technician can repair inside an airport once they die, you're probably looking at having to spend some money to replace the unit.
This sounds suspiciously like a hardware fault. If you are in an area with a Genius Bar or an authorized Apple repair centre, I would take the unit in for testing/diagnosis. As these are always-on devices, they should be able to sustain brownouts and outages without issue.