Here are your parts
Replacement parts for WHIRLPOOL KEYS677EQ0 | AppliancePartsPros.com

Here is the wiring diagram
http://www.servicematters.com/d...0Sheet%20-%203401725.pdf

My dryer vents straight up through the roof. What is the best way to clean a vent like that? Rerouting the vent is not an option.
Sometimes it is just the screen at the vent that is clogged up or it may be at a 90 degree fitting.
Most often it must be cleaned from the top and at minimum the screen/wire mesh must be removed.
Have heard of people using a chain with a couple heavy locks on it to clear a clog or use a plumbing snake.
Also some folks have had luck with a leaf blower or a compressor.
In any case all methods are a pain as most roof vents are difficult to remove for cleaning.

Should I be checking the thermostats and the thermal cutoff?
Try the following first, then you will know if it is a dryer or a vent problem.
Undo the dryer from the exhaust
Run a load
If the load dries correctly then you have a vent problem.
Clean/check the vent system. Make sure the outside louvers open fully.

If you do not want lint in the house.
Take a pair of pantyhose, put one leg into the other and attach this to the dryer exhaust.
Leave enough room behind the dryer so it can blow out like a windsock.
Run a load.

Both the above will allow you to check the air flow and temp.
Also the above is NOT recommended for a gas dryer due to possible carbon monoxide problems.

Could be the thermal fuse Item 6 in Section 3 has blown.

Easiest way to check is with a meter. Remember to unplug the unit and also disconnect one side of the fuse when measuring it so that you do not read an alternate/parallel circuit path.

Jiggery Pokery method is to unplug the unit, short the two wires together, tape them up so they cannot short to anything and give it a try.
If it is blown replace it ASAP as this is a critical safety device to prevent lint fires/ house fires.

Strange that it should blow after just some basic cleaning.
Check that lint has not clogged up the blower wheel due to moving the unit.
Also check that the blower wheel is OK and is not slipping on the motor shaft, though uncommon sometimes the thread can strip and they slip on the shaft.
Also could be that moving the unit caused lint to settle and clogged the vent solid.
Note: that the most common cause of it blowing is a badly clogged vent system.