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If you look under the sink and there are two wires running too the underneath of the tap, it's micro switch operated, which I think it is because the other taps work OK. Another way to check is open the kitchen tap fully, then very carefully open the bath room tap, so as not to let any water out, kitchen tap will run when you trip the bathroom micro switch.
I found them easy to replace. just take the tap handle off by undoing the grub screw under the hot/cold indicator and look underneath the cover.