After some internet research I came across a service manual for my model of washing machine, a Whirlpool Calypso. The manual had a nice troubleshooting guide showed me how to check the control panel with a multimeter to see if there was a button out. The manual also showed how to get access to the control panel (shove a putty knife in the crevice where the control panel meets the body of the washing machine and pop the spring clips).
I ohmed out the control panel according the to the service manual and sure enough it was bad. I ordered a new one from Appliance Parts Pros. It was kind of expensive, but much cheaper than a new washer so I decided to roll the dice and order it.
The service manual was really good about having procedures for testing things and had procedures for replacing many components like the control boards and motors and things but it didn't have a procedure for replacing the control panel.
The hardest part was removing the old panel. First I had to wrestle witht he plastic end caps that bookend the control panel. Then I found that the bar which holds up the circuit boards you can see in the picture below was glued to the back of the control panel. I pulled out my trusty putty knife again and used it to pry the control panel free. Luckily it wasn't very strong glue and it came free pretty easily once I got it started with the putty knife.
Once I had the old panel removed putting the new one in was as simple as putting the end caps back on, connecting a ribbon cable to the control board, and reversing the rest of the disassembly.
And luckily the washing machine started working again as soon as I re-connected power.