G'day Bondy,
The magneto wiring is not required for you to be able to use the electric start. The magneto wiring is simply used to stop the engine- and we can do that with the switch that Honda has provided that is operated by the throttle lever.
Of course you'll need to find out why it's not working at the moment. I have the strong suspicion that this problem is related to the fact that is seems the choke isn't working either with the problem being that your not getting full travel of the throttle arm.
Once your electric start is up and working and the throttle travel thing is sorted you'll start the machine by placing the throttle to the choke position and then turning the key.
To stop the machine at the end of a hard days mowing you'll simply place the throttle lever to the stop position. Turning the key will have no effect here.
To confirm the throttle arm fault is caused by lack of cable travel, remove the cable completely from the engine and see that the throttle lever on the engine closes the choke in the start position and makes the magneto switch in the closed position.