Diagnosis
The physical mechanical power switch has failed. It's a separate component from the touch-sensitive thermostat and timer buttons. Debris or a broken internal mechanism is the culprit.
How to Fix It
This requires opening the unit. After unplugging, remove the screws. The power switch is mounted on the front control panel. You can try spraying a small amount of electronic contact cleaner (not WD-40) into the switch mechanism and working it back and forth to free it. If the plastic rocker is physically broken, the switch needs to be desoldered and replaced with an identical momentary switch.
If That Didn't Work
If the switch seems fine physically but the heater still won't power on via the button, the issue could be on the main control board where the switch connects. A broken solder joint on the board where the switch attaches is a possibility.
Heads Up
When working on the control panel, be extremely gentle with the ribbon cable that connects it to the main board. That cable is easy to tear and very difficult to replace.