Diagnosis
The power light is on a separate circuit from the heating function. If the light is on but nothing happens when you turn the thermostat knob, the issue is almost certainly in the thermostat circuit or its connection to the heating element.
How to Fix It
Unplug the heater and open it up. 1) Locate the thermostat—it’s the round component on the control panel that the knob attaches to. 2) You’ll see two or three wires connected to it with spade terminals. 3) Gently pull off each wire, one at a time, and check the terminals on the thermostat and the wire connectors for corrosion or a loose fit. Clean them with a bit of sandpaper or a wire brush if needed, and reconnect them firmly. 4) Plug the heater back in (be careful with exposed internals) and carefully try turning the knob. You might hear a faint click from the thermostat engaging.
If That Didn't Work
The thermostat itself may have failed open. You can test it with a multimeter for continuity while slowly turning the knob. It should click and show continuity at some point in the rotation. If it never does, the thermostat is dead and needs replacement.
Heads Up
When testing a plugged-in heater with the case open, keep your hands away from all internal components and do not touch any metal parts with tools. It’s best to have a friend help by plugging/unplugging while you observe.