The 665 tells us that the unit was made by Whirlpool but there should be about 8 digits after it which is specific model info.

Not sure if the Serial Number you show are the rest of the model number, please check it.

Here are your parts
Replacement parts for KENMORE 66515769000 | AppliancePartsPros.com

The Sears parts site should have a parts breakdown.

See the attachment for the tech sheet.

Sounds like a control panel (keypad) or a control board problem.
As you can see in the tech sheet a number of keys share common lines so if a certain set of keys do not work then that is where your problem is.
This still does not narrow it down enough so the usual way to troubleshoot is to check the keypad with a meter, if it checks out OK then replace the control board.
Still a bit of a crap shoot but your odds are better.

Also you may want to 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.
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.