I'm thinking the 555 chip inside has died, so I might swap it out and see if it puts me back on track.
You wont find a 555 in the relay. It is the ECU that stops the fuel pump running after a few seconds when the ignition is turned on.
A K-Jet is ECU-less, the 555 in the relay is what controls prime and pulse.
So, if there is a 555 in the fuel pump relay to stop the current after say 2 seconds, then what makes the fuel pump run continuously when the engine is running.
The pump takes about 2 secs to pressurise the fuel circuit, when 'up to pressure' it stops, it is not specifically 'time' dependant. As soon as you start the engine it takes fuel from the system and the pressure drops so the pump has to keep going to keep up the pressure in the system. If you disconnect the fuel feed into the metering unit and turn the key it will just keep on pumping rather than stopping after 2 (ish) secs.