The given model number is missing several digits at the end.
Please check it.

Here are your parts using the model you gave. You will have to choose the correct unit from the Matching Model Numbers section.
Parts for ED25RQ - AppliancePartsPros.com

I would need a complete model number to get a wiring diagram for this unit.

I am assuming that the interior lights work.

Try turning the thermostat off and back on to see if it will fire up.
If it starts working replace the thermostat.

You may be stuck in a defrost cycle.
Try turning the defrost cam if this unit uses a defrost timer. The defrost timer is probably in the control console, there is usually a hole to let you turn the cam without having to disassemble anything. the cam only turns in one direction.

If it fires up replace the timer.

Here are your parts
Parts for Whirlpool ED25RQXYW01 Refrigerator - AppliancePartsPros.com

You are correct, this unit does not have a defrost timer.
It uses an adaptive defrost board (item 6 in the Control Parts section).

See the attachment for the tech sheet.
It shows how to force a defrost cycle.

I would replace the defrost thermostat.
Note that I am not a fan of immersing thermostats in water. If any water gets into the thermostat it will freeze and screw the thermostat.
You are better off holding the face of the thermostat onto an ice cube till it gets cold enough.

While in that area also check the defrost heater, should be around 20 ohms.

If the heater is OK then I would replace the defrost board.
A bad defrost thermostat will not lock the unit up in a defrost cycle.