31313, 31570, and 31569 should all provide basic functions. When you assign 31313 to AUD, does the remote blink twice in response? Does the LED blink when you press Vol +/-?.

Many of the functions on the Kenwood don't have obvious places to go on the remote. Once you get volume and power working, we can tell you how to assign 5 digit EFCs to button to get functions like discrete inputs or Bass Boost working.

Try this experiment. With 31313 assigned to AUD, pres AUD. Then Press P, followed by the digits 60803. When you press the last digit (3), the power should toggle.