Kenmore Elite Dryer....Push Button...doesn't stay on

Most likely it is dirty and lint deposits have built up, as it always does, in all the air ducts which will cause the main safety to blow. A little vacuuming out does nothing. You will have to take it apart. You will have to clean the drier out thoroughly or it is all for nothing. Every air duct will have to be removed and scraped and washed out to remove baked on lint deposits which bakes on like a concrete lining. This will burn so the manufacturer has installed the safeties to keep from having a house fire. The safeties are oval-shaped with a round center part containing the safety with two plug-on terminals for the wires and when they burn open, from the dryer being dirty, nothing will work. Using an ohm meter you put a lead on each terminal of the safety, with one wire unplugged, and if it is good you will have a reading of about zero; if it is bad it will read infinity. Driers were the number one cause of fires at one time, hence now we have safety fuses. Look at: http://www.youtube.com/watch?v=MMP7BW1lLs4 to see how. Also: http://www.youtube.com/watch?v=iBiRw-KElC4 ; http://www.youtube.com/watch?v=yFTZGgVisCI
http://www.electrical-forensics.com/Dryer/ElectricClothesDryers.html
http://fixitnow.com/wp/2010/10/09/whirlpool-dryer-model-le5800xsw3-wiring-diagram/
When you finish cleaning the drier it is best to also replace the belt, too, because you do not want to have to disassemble it all over again and if the belt is over two years old you should replace it.

Note that on most of the videos it does not show them washing out the parts, but this is important because it's the only way to really get the parts clean, just don't get motor wet or the capacitor if it has one. Vacuum out the motor to remove all dust. I would move the drier out to your garage to have plenty of space to work. You also want to check your vent and clean it, also. The vent should have a short run and be straight as possible. The job is really easier than it sounds. There is no shortcut that will work; it has to be cleaned well. Before you re-assemble it make sure to use white lithium grease (high-temp grease) and grease all bearings. Just vacuuming out the dryer a little does little good. DrTom4444