If you recall manual chokes, you pulled a wire which in turn pulled on a lever/cam on the carburetter, this turned the spindle fitted with butterfly flap valve, partially closing this valve choked the air supply resulting in a richer mixture to enable the engine to start. As the engine warmed up it was up to you to release the choke bit at a time.
A automatic choke does the same basic function, on a cold engine the valve is already in the choked position, Heat from the circulating water is used in some form of thermocouple to make the linkage move to open the valve which should be fully open when the engine has reached operating temperture.
So have a look at the carburetter while someone presses the accelerator pedal, where you see movement of the accelerator cable thats where you will find the choke linking rod between carburetter and choke housing.
The choke housing will probably be aluminium, not more than 3 ins. in any direction ( I would think) and will have two pipes for the flow and return of water, although it could be mounted some other way where you cannot see the pipes. Probabley have circular cover on one side.
You could buy kits to change from automatic to manual at one time.
Having said that, it sounds as though yours is working in principle but needs tweaking