Q: Status light fails to light even with power connected.  What could be the problem?

The power supply is dead.. or the board inside the Airport is dead.

Tell us exactly which model it is .. the label on the bottom or molded into the rubber pad.. A1xxx

This is very typical failure on the Time Capsule A1409 and perhaps the equivalent Airport Extreme A1408.

There is a fix .. but for older models may not be worth it.

If you just updated the firmware and that triggered the failure.. it wasn't the upgrade itself .. but the extra stress on the TC brought the problem forward.

See fix here. https://sites.google.com/site/lapastenague/a-deconstruction-of-routers-and-modem s/apple-time-capsule-repair/gen4-a1409-i…

I think this will also work for the Gen5 AE.. if you have earlier ones it is dead power supply and is typical issue.