Here is the parts breakdown
Replacement parts for WHIRLPOOL LER8648PW1 29"electric Dryer | AppliancePartsPros.com

And a wiring diagram
http://www.servicematters.com/docs/wiring/Wiring%20Sheet%20-%208576811.pdf

First try flipping the breaker off/on slowly a couple times. Sometimes you can loose half the line without actually tripping the breaker and the heating coil requires the full 240 volts.
If this does nothing.
Measure the voltage at the plug
L1 to L2 should be 240 volts
L1 to Neutral and L2 to Neutral, both should be 120 volts.
If OK
Unplug the unit and check the wires at the unit's terminal strip to ensure they are properly connected and none of them have burned off
If OK
Plug the unit in and check the voltage at the terminal strip. This is just in case you have a bad line cord. Be careful 240 volts is lethal.
If OK
Unplug the unit and check the heating coil and thermostats etc. for continuity.
Heating coil, should be 10 ohms approximately.
Thermostats and thermal fuse all should be 0 ohms.
If the heater fuse (on the heater assembly) is blown then both it and the hi-limit should be replaced, see below additional info.

Be sure to disconnect one side of and device you are measuring this prevents reading an alternate/parallel circuit path. Also use your most sensitive ohms scale.
There is a good Sticky at the beginning of this forum on meter usage.

Blown Heater Fuse
Unplug the unit and check it with a meter, should be 0 ohms.
Be sure to unplug one side of it when checking it.

If it is blown you have to find out what caused it to go.
Note: that sometimes they do just blow on their own but changing it without checking other things is a gamble.

Check the heating coil.
The old heating coil may have had this problem but does not hurt to check the new one. Would not be the first time a replacement part was bad.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case. This can cause it to run on high and the thermostats cannot regulate it.

If the above is OK then you will also have to replace the hi-limit as it should have regulated the temperature so the fuse did not blow.

You still have to find out why it blew.
Check that the belt is OK.
Check the seals (drum etc) in the unit. The air is pulled over the heating coils, through the drum and pushed out the exhaust. So any large seal leak will pull in room air and the cycling thermostat on the blower will run the unit hot.
Check that the lint filter is not coated with fabric softener residue which greatly reduces air flow.
Check/clean your vent system.
Check/clean the blower wheel.

If all OK you may want to replace the cycling thermostat as it's contacts may not be opening (welded shut).