Here is the tech sheet
http://www.servicematters.com/d...0Sheet%20-%208572216.pdf

Try running the diagnostics to see if it will clear.
Next, I would check out the door switches.

I would then check the heater, the hi-limit thermostat and the thermistor.
Though this is a long shot because I get the impression from you post that it gives the error without running a cycle.

Also it is troubling that the new board ran for a week and then messed up, seems like something else in the machine took the board out so I would check all components with a meter.

The DW runs for a few days after by initiating the service diagnostic cycle. But it goes through the diagnostic cycle without a hang up. My questions:

1. Is it right to assume that the problem is intermittent because the DW will go through several cycles before it gets hung up?
Yes. It will make it a pain to repair

2. The sensing light does not go on (except when all of the lights are on). We thought it did go on before. What part would need to be replaced if that identifies the problem?
Does not really point to a specific part

3. Is there a simple way to check the heater circuit?
Yes with a meter you can check the thermistor, the element and the hi-limit thermostat. They will probably all check OK since your problem is intermittent.

4. Is there a booklet for this model on repairs?
I can tell you that it is a Whirlpool unit but I have no way to cross reference the Kenmore number.

Since it is an intermittent problem the most likely cause is a loose or bad connection. I would check the heater and the thermistor connects. The door switches also but this is a bit of a long shot.

You will have to see if they fit snugly, that the pins/connectors are properly crimped and that they are clean. Also check the wires that go from the control board to under the unit.

I would also try the following.
Unplug the unit
Open it up
Clean the connector on the board and the connectors themselves. 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.