The model number K301AQS; the spec number: 47640; the serial number: 15098858.
I couldn't find the DOM, we guessed the model year is about 1986.
Okay, that will be the first thing to determine.
You're going to need an engine manual. Fortunately, if you go to http://www.asberry.net/Kohler.htm you can download one.
We know that an engine needs three things to run, compression, fuel and ignition and all at the proper time.
By using ether, you've effectively eliminated fuel as the problem. So first I would check for spark. Pull the spark plug, hook the plug wire back up and using insulated pliers, hold the metal part of the plug against the engine head while someone tries to start the engine. If there's a spark, then I would check the compression with a compression gauge. It's hard to determine the actual compression because of the compression release, but it will give you an idea.
Also, you can have ignition, but if it's not delivered at the right time, then the engine won't run. You'll need to determine which ignition system you have. You could have a points issue, a problem with the pushrod for the points, or for the electronic ignitions a flywheel that's moved out of position.
Check these things and let me know what you find and we'll go from there. Thanks, PK.