As Star mentioned, the oil leak will probably be from the gasket that has gone hard and is now leaking
The design is not the best and made worse by people over tightening the cover bolts to the point they deform the steel cover
They should be tightened to 8 ft lbs so the gasket is just compressed within it's springy range
Usually they are chopped up
I tighten the finger tight by rolling the deep socket between my fingers till it stops then 1/4 turn
If it leaks then another 1/4 turn .
And that is after I have knocked the bolt holes flat again .
As Slomo mentioned the fuel will probably have gone stale by now and the carb will most likely have some corrosion clogging up the jets
Clean the carb as per the instructions
Here
Go to Kohler and download the service manual for your engine and adjust the valve lash as per the instructions
Replace the gasket, Prime Line , Rotary, Oregon & Stens all do aftermarket gaskets & the Prime line ones seem to be better .
In fact I use mostly Prime Line parts for Kohler engines