AirPort Extreme won't turn on after power outage
Solution 1:
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.