I have a Kenmore Elite dishwasher, Model No. 665.17462300, with a broken heating element. Sears claims there is no replacement for this element, yet I've found websites that claim that the element can be replaced with an OEM part, no. 8194200 that has a new electronic module as well as the coil for around $150. Can you replace just the heating element without the electronic board or do they both need to be replaced? Will this OEM part work with this dishwasher?

The element is easily replaced by itself. Look at these: https://dl.dropboxusercontent.com/u/...Dishwasher.pdf
https://dl.dropboxusercontent.com/u/...e%20Manual.pdf
https://dl.dropboxusercontent.com/u/...Electronic.pdf
You will need to use a meter if you need to test other things. Make sure no fuses are blown. Sometimes you have fusible links and fusible resistors soldered on the circuit board, but they are simple to replace with a soldering iron and a meter. I doubt that you will have any problems, but if the terminal ends are burned and need replacing you MUST replace them with hi-temp terminals that are chrome. Other than that it's no problem.