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Replacement parts for Kenmore 66516462300 | AppliancePartsPros.com

See the attachment for the tech sheet.

Check out it's troubleshooting section.

Most likely culprits are the keypad (AP3606065) (available in different colors) or the control board (AP3775506)

Check the pin out matrix for the keypad, see the tech sheet.
As you can see keys share wires/tracks so you are trying to narrow down the problem using this.
For instance the Start button uses P1-12 but so do Delay Hours, Rinse Only, Quick Wash and Cancel. So it these keys also do not work then P1-12 is a likely cause.

The usual way to troubleshoot is to check the keaypad outputs with a meter. If it is OK then replace the control board. Still a bit of a crap shoot but your odds are better.

I would try the following.
Connector Cleaning / Checking
Unplug the unit
Open it up
Remove the ribbon cable from the keypad to the control board at the control board.
Clean the edge connector. Use electronics contact cleaner or isopropyl alcohol.
If it is the type where there is not a connector just the insulation is removed, it can be cleaned with a pencil eraser. Do not use an ink eraser as it is too abrasive.
Check the tracks in the ribbon cable for hairline cracks. A magnifying glass helps here.
Would not hurt to clean the connector on the board. Use electronics contact cleaner or isopropyl alcohol and a tooth brush. Be gentle and do not reconnect till all the solvent evaporates.
Reconnect everything and give it a try.