98 XJ8 fan running on after engine switch off

[Euro5stars]

The cooling fan has just started doing this from the weekend. Temp gauge is plum in the middle after couple of miles and didn't run on when the engine was switched off. However, after a longer trip of about 8 miles, the fan runs on for 5-10 mins. If it's sat in traffic then you can hear it running with the window down. Some forums suggest thermostat, some suggest the coolant temp sensor (bit of a soft question, but what's the difference??) and others say the water pump and hoses may not be shifting coolant around quick enough. Jaguarsteve has mentioned to check the water pump for swirling and gradually raising the revs (on previous posts), to see movement and it seems ok.
Is the thermostat in the black dome with a lead coming off the top near to the coolant reservoir? Think the water pump is closeby to that, but where would the coolant temp sensor be?

[P700DEE]

My money is on the thermostat, they fail closed and you really don't want it to over heat ! Check the water pump , early ones were prone to loosing their blades but I doubt a 98 car is on original thermostat and pump.

[Euro5stars]

Resurrection time.
Found the problem down to the impellor in the pump falling apart. Mechanic showed me what was left. Only two vanes left in the pump and the top crown flapping around. Does the fact that the rest of the plastic vanes are missing mean any long term issues within the coolant? Any suggestions on the benefits of trying to flush the vanes out?
Cheers