We are going to need the full engine number.
If you have one of the SAM type ignitions with a powered ignition timing control then it is most likely not getting power.
I have a fleet of hire cars and they have the same problem, when cranking the voltage to the ignition unit drops to 10V and they won't fire.
The exact same engine with points fires right up while cranking
If this problem has only started happening since you changed the ignition switch, then the problem will be in the switch.
The power to the ignition might be coming from the alternator output which is 0 till the engine fires on some configurations
Otherwise the ignition is self contained and is controlled by grounding the module in which case it has a separate circuit inside the switch,
M + G = off
M + nothing = on
Letters are on the back of the switch
However before we start guessing we need more information.
1) full model number from engine
2) what type of ignition ( the SAM type will have more than 1 small wire going to the coils )
3) a full service manual for yo engine is available from Kohler and from
https://mymowerparts.com/about_us.php