Is there no id label on the engine with a spec number? John Deere does often specify parts from their vendors, such as Kohler and Tuff-Torq to not have any ID numbers that will cross-reference and this does cause quite a bit of trouble. Usually the engine will still have a label on the side of the block with the exact model and spec number though.
Without the proper spec number, you're probably limited to taking the old coil into the JD dealer and having them match it.
With the full model and spec number, the proper Kohler coil number can be found.