My dryer will not start. When I try to start it or select the setting to make it start, it just makes a humming sound.

Could be a bad motor or maybe as simple as a broken belt.

If you can hear a constant humming and it heats up, you probably have a broken belt.

Since it has run long enough to wear out the belt it needs cleaning, too. You need to totally tear it down and scrape out and wash all of the parts except the motor, which you vacuum out. You can go to YouTube to see videos on how to take it apart, but do a very good job or it will fail again quickly. Get some white lithium grease and grease any bearings you find. You need to clean out a drier every two years, at least, according to the manuals.
Look at: How to clean your clothes dryer. How to make it hot again. - YouTube to see how to tear it down. Also: Mr. Appliance Tips: How to Clean Your Dryer - YouTube ; Cleaning a Dryer Vent and Lint Trap - YouTube
Electric Clothes Dryers
Whirlpool Dryer Model LE5800XSW3 Wiring Diagram | Dryer Repair | Fixitnow.com Samurai Appliance Repair Man
The job is very easy for anyone to do. Just get it out into your garage or somewhere you will have plenty of space. Pop the top and unplug the wiring harness, then you can see where the screws are that hold on the front. You will need to unplug the door switch. You should lay it on its back to make it easier to work on before pulling off front. Make sure to clean it very well and don't miss the duct on the door going to filter or the other ducts in the rear, and the blower wheel and housing which is very, very important. A clog of cement-like lint build up can hide anywhere. Don't be afraid to take apart anything to be able to look inside and clean it. One missed place will necessitate doing all this again and no one wants to redo something. Grease all support rollers with white lithium grease and reassemble and you have a new dryer.
Sometimes, it will not start due to a safety being blown, but it sounds like you have a broken belt. The safeties are round with an oval flange where two screws hold it on the duct. You need a volt/ohm meter to find any bad safeties.
Here are some manuals: https://dl.dropbox.com/u/8010163/GE%...%20Manuals.zip
https://dl.dropbox.com/u/8010163/How...e%20Manual.pdf