First of all, you don't need the AirPort Extreme to "show up" to completely reset the device.
You will want to do a Factory Default Reset (A Hard Reset will not completely wipe the stored network settings on the AirPort)
Power off the AirPort Extreme by pulling the power supply cord from the back of the unit
Wait 5-10 seconds
Hold in the reset button first, and keep holding it in while you plug the power cord back into the AirPort Extreme
Keep holding the reset button for another 7-8 seconds after you plug the power cord back in
Release the reset button
The AirPort Extreme will display a solid amber light for a few seconds then the light will turn to flashing amber
You are done. You can refer to Apple's Support document on resets if you need more info:
How to reset your AirPort base station - Apple Support