Almost certainly all you had to do was toggle the Accessibility permissions off and back on again.
This has been an annoying bug with OS X since Mojave, and continues in Catalina (as well as Catalina’s new twist of sometimes not even allowing the checkbox to turn on).
Unchecking and rechecking the boxes wasn't enough.
The permissions had to be deleted with the "-" button, and then re-added when KM made an attempt to perform a control action. Then the KM Engine was re-enabled manually via the warning notification in its menu bar menu.
Turns out it wasn't just KM either - BetterSnapTool had the same issue. I didn't have to uninstall it so that suggests I didn't have to uninstall KM either - like Peter said