DIAGNOSIS: If the power brick is warm but the router is completely dead, the brick might still be getting current, but the router isn't receiving it. This points to a failed internal power supply in the router or, less likely, a faulty power brick that's not outputting the right voltage anymore.
FIX: 1) Unplug the power brick from the wall and from the router. 2) Check the cable running from the brick to the router for any kinks, cuts, or damage. 3) Firmly reconnect the cable to both the brick and the router. 4) Plug the brick into a different wall outlet, preferably one you know works. Listen for a faint electrical hum from the brick and feel if it gets warmer. If the router lights still don't come on, the router's internal hardware has likely failed.
IF THAT DIDN'T WORK: If you have a multimeter, you can check the power brick's output. It should be 12 volts DC. If it's significantly lower or zero, the brick is bad. Unfortunately, without another compatible 12V power supply to test with, it's hard to isolate the issue.
HEADS UP: Before you conclude it's dead, try a hard reset. With the router unplugged, press and hold the recessed reset button on the back with a paperclip. Keep holding it as you plug the power back in, and continue holding for another 15 seconds. If any life was left, this might jolt it back. If not, it's time to contact Netgear support.