There is a point when it doesn't save you anything when trying to fix an older version of the device.
In this case these older AirPort Extreme's WiFi services are somewhat dated (802.11a/b). Today most systems are 802.11n or now 802.11ac which is the new super fast standard that came out this summer.
The other factor here is the WiFi security services. I don't believe you can setup WPA which is the stronger encryption Vs WEP which is very easily cracked o these older units.
The bottom line here is I think its' time for a new one. And yes, it does sound like the power supply is gone.