:welcome:
With a "flat" battery , measure the battery voltage then jump start the mower.
Measure the battery voltage again, engine running,
If you see 13V to !5V the charging system is OK
If not go to the engine and measure the voltage between the rectifier ( Red or Purple wire ) and ground
13V to 15 V charging system is OK & you need to clean all of the wiring terminals , particularly the grounds.
If voltage is lower then check the voltage between the two same coloured wires which is the AC FEED
You are looking for AC Voltage of 28 V AC ( +14 V & - 14V = 28 V AC ) to 40V AC
AC good, DC bad = dead rectifier, again clean the ground contact and double check your readings.
This is general information.
Anything more detailed will need the engine numbers from the Kawaka.
Get back with this & we can go through how to verify the potentailed failed parts.