Airport Extreme

This additional information indicates that the internal power supply may be having problems, so replacing the outboard power supply would likely not help in this situation.

But, if the seller agrees to take back the outboard replacement power supply for credit if it does not help, then your risk is minimal.

On another note, an older version of the AirPort Extreme is at least 5+ years old and as much as 9+ depending on when the device was originally placed into service.....and the average useful life of a router designed for home use is about 5 years.....so if it were me, I would replace the older AirPort Extreme with a new router.

If the AirPort Extreme operates OK otherwise, then the LED has failed.

If the AirPort Extreme is the "tall" or "tower" version and it does not operate at all, then the AirPort Extreme has failed and will need to be replaced.

If the AirPort Extreme is an older "flat" or "square" version and it does not operate at all, a new outboard power supply might bring new life to the AirPort Extreme.

Original OEM Apple AirPort Extreme Base Station Power Supply AC Adapter A1202 U | eBay

Is your Airport the Gen5 model?

It will have A1408 marked on the label as model number.

They are subject to board breakdown like the previous model.. I doubt it will be the external supply but you can use any plug pack.. if you have 12-16v 2A or more suppply give it a try.

The oven bake solder reflow method does also work on the Gen5 so it might be worth considering. It did fix another AE gen5 here..

See the instructions for the equivalent model TC (gen4)

Gen4 A1409 issues - LaPastenague

I am trying to write up the instructions for the Gen5 AE but i had a bit of a meltdown.. got the board too hot and ruined it. The oven temperature must be controlled really well. (my cheap little toaster oven went into overdrive).