Kohler offers many free and downloadable engine manuals with parts schematics at their web site. Exceptions may occur if mower came from giant chain store, and is proprietary to giant store specs.
To get the actual part, aftermarket mail order is usually cheaper than OEM. "Kohler [part number]" to Google or Froogle will likely give you many sourcing choices.
If your mower is missing the data plate and you can't find an engine number, there are a multitude of web forums devoted to small engine care and feeding. Some posted photos of your unit will very likely produce a means to finding the right part.
Note - if the engine's data panel is of an adhesive design, (and it probably is) you might consider a method to preserve or record the data, as these adhesive "plates" have a tendency to disappear before the engine resigns. I have our dishwasher and GMC pickup data plate / parts codes decals skewered onto my shop wall with thumb tacks. Both were swinging in the wind when I removed them from their attachment.
Workaround to keep the hay at bay, if procuring the correct plug proves a lengthy process:
Change the oil on a level surface and replenish unit with Kohler manual suggested quantity of oil, in the appropriate viscosity. Make a note of new oil's visual oil level, say x-threads down from the top. Get a substitute plug locally from a lawn mower shop. If the replacement has a dipstick, you are fine as long as stick portion does not bottom out. Bending a too long stick in half will not hurt anything, as long as attachment of stick to plug remains sound. Plugs usually cost no more than $6 or $8 US.
Note that the plug could be pipe thread, which has a taper. If you find a hardware store substitute, you'll also need a gasket (or large O ring) to keep the oil in.