There is a solenoid on KIOTI tractors to shut them down. There are two types. LK's had an external solenoid type with a timer under dash to pull the solenoid for a few seconds to stop/shutdown the engine.
Key switch could be the problem. Remove and replace with new. If it doesn't fix proceed to below.
Go to right side of engine at injection pump, find solenoid, find single wire, unplug, take a jumper wire to 12V+ side of battery and touch (dont hold) to see if it moves. If it doesn't move, solenoid is bad. If it does timer is bad. NOTE: timer went bad for one reason or another. Buy both new timer AND solenoid. It would be bad if you hook up new timer and existing solenoid is drawing to many amps and burns up NEW timer.
Most dealers stock these parts. Call if you cannot find at your local dealer.