John - Proper nomenclature for that item is "afterfire control solenoid" - it controls fuel flow to the power circuit only inside the carb. To test is a 2 step process... 1) when you turn ign key to run, solenoid should energize with a sharp audible "click"; 2) when removed from carb body, spring-loaded plunger should move freely with no sticking when depressed by hand.
Beyond all that - IF you now have installed a NEW genuine Kohler fuel filter specifically for that engine in the correct flow direction and have fuel flow up to the carb itself, you have a major decision to make: Either to do some serious detailed repair yourself (like a carb teardown/rebuild) or take the machine to a shop.