Kohler toilets are excellent products. They often are marked on both the tank and the lid with part numbers. The tank will often have four lines of numbers, and one of those lines is the part number, usually in the format Kxxxx where the x's represent numbers. The other three lines include a date code and plant/manufacturing/lot codes.
Depending on the age of the toilet, there may or may not be a sticker on the underside of the lid. If there is a sticker, the part number on the sticker is actually just the sticker's part number, not the tank lid part number. They often use the same sticker on different tank lids, so you will need to remove the sticker to reveal if it is stamped with a tank lid part number.
Kohler products went through a transition at some point. They used to use a number similar to the tank, Kxxxx. Eventually they changed their tank lids to use a number that starts with an 8. Many products were then given the original K number and the new 8 number. In most cases if there is a new and an old number listed on this site, the old number was the one you are most likely to find on the toilet tank lid, and the new number was one that they used in their spec sheets. Now they use 10xxxxxx numbers as well for toilet tank lids.