Here are your parts
Replacement parts for WHIRLPOOL LER5636EQ2 | AppliancePartsPros.com

Here is the wiring diagram
http://www.servicematters.com/d...0Sheet%20-%203406688.pdf

My dryer stopped working after a long day of use.
I am assuming that it will not start.

I opened the back panel to check all of the connections, and found that the rubber sheathing around the resistor was black and split and wanted to know if this could be the cause of the dryer not working anymore.
No. This resistor is just used in Auto cycles when the heater is shut off the timer gets power through the heater and the resistor reduces the 240 volt heater voltage to 120 volts for the timer motor. But I would check it as it sounds like it has been overheating.

First flip your breaker off/on slowly a couple times. Sometimes you can loose half the line without tripping the breaker. Note the drum light will not work if you lost the half of the line that runs the motor.
Better yet check it at the plug
L1 to L2 should be 240 volts
L1 to Neutral and L2 to Neutral, both should be 120 volts.
Also check the wires at the unit's terminal to be sure one of them has not burned off.

Next would be the thermal fuse on the blower case.
Easiest way is to:
Unplug the unit.
Remove one wire from the fuse
Measure it with a meter, should be 0 ohms.

Jiggery Pokery method is:
Unplug the unit
Remove both wires from the fuse and short them together.
Tape them up so they cannot short to the frame.
Plug the unit in and give it a try.

If it is blown replace it ASAP as it is a critical safety device.
Also clean/check your vent system as this is the most common cause of this fuse blowing.

Give the interior including the blower wheel a good cleaning.