James, from what you originally told us it could simply be a problem with the electrical switching of the Kitchen tap. As you say the toilet work fine on its own and both work fine with the toilet switched on then the pump and water system are not at fault. Its the kitchen tap is not able to energise the pump properly; almost certain the taps micro-switch, but possible the wiring to it or the associated connectors. First wiggle the connectors if that does not cure it ; With the tap open, short out the taps two wires ,one to the other. If it then works its the microswitch; you might be lucky and drive any damp out of it with just a hint of WD 40. Go careful it leaves a deadfull stink and SWMBO will not know if to curse or thank you!