Here are your parts
Replacement parts for WHIRLPOOL GEW9200LW0 DRYER | AppliancePartsPros.com

See the attachment for the tech sheet.

And here is a service manual
L-69 Whirlpool Duet Gas & Electric Dryers Service Manual - ApplianceDigest.com

You have to join the site to download the manual but it is free and this is a very good site for manuals and other info for the appliance repair DIYer.

In the tech sheet wiring diagram you will see that in the motor circuit there is a belt switch which opens when the belt breaks and a thermal fuse which opens when the unit overheats.
It could be one of these is the problem. The click you hear may be a relay on the control board trying to start the motor.

If it is the thermal fuse be sure to check/clean your vent system as it is the most common cause of the fuse blowing.

I cannot say if you can do this it depends on your mechanical ability but it is not that difficult.

You will not be able to tell if the fuse is blown by just looking at it so:
Unplug the unit and check the fuse with a meter should be 0 ohms.

Jiggery Pokery method is.
Unplug the unit.
Remove both wires from the fuse, short them together and tape them up so they will not short to the frame etc.
Plug the unit in and give it a try.

If blown, replace the fuse ASAP as it is a critical safety device re: prevents lint fires which can turn into house fires.

Also clean/check your vent system and the blower as these are the most common causes of it blowing.

It could be something else but I would start with the above at least that is my best guess.