First things first How old is this unit? If it is less than a year Kenwood will replace it. Now if it is older than a year have you verified that you have 120Vac from the wall outlet? If you have we can proceed with troubleshooting the unit. Any of the following will require some disassembly of the unit. Again I want to stress that If it is under warranty opening the unit may void the warranty so please be aware of this. We need to determine the point of failure. It could be the switch, a fuse or possibly a bi-metal thermostat has blown open.
We will start with the fuse. Start by unlpugging the unit and removing the housing. At the entry point of the AC line there should be a fuse holder. Remove the fuse, if it is a clear glass fuse check the fuseable link inside has it been blown or not? If it is not a glass fuse you will need a ohm meter or continuity meter. If blown replace with correct amperage fuse.
Now for the switch, locate the power swith on the inside. Take a volt meter and measure from ground to the switched side of the contacts. When open you should get no voltage, when closed you will read 120VAC. ( BE CAREFUL THIS IS A SHOCK HAZARD)
Bi-Metal Thermostat.
Locate something that looks like this.
http://img.directindustry.com/images_di/photo-g/bimetal-thermostat-120372.jpg
If you cannot find one there may not be one. This is for overheating purposes.
Lastly here is the manual for your mixer stating the one year warranty. Good luck!
http://www.kenwoodworld.com/Global/Instruction%20books%20(User%20manuals)/Food%20Mixers/88907iss3%20KM001%20to%20KM006%20English%20only.pdf