The basic troubleshooting is just like any other toilet with the concerns whether the fill valve is filling the tank to the proper level, fully shutting off when the proper level is reached, and not leaking spraying water everywhere.
If you have one of the toilets where support was discontinued you should check with Kohler to see if they are still offering a credit towards the purchase of a new Kohler toilet when your toilet dies. It’s one thing to have toilets with expensive and hard to get parts but when a manufacturer drops support for a product such as the older Pressure Lite Models such as the one-piece San Raphael & Rialto Pressure Lite Toilets and the San Miguel K-3406-U-EBU, you are left with a toilet that is only good until it needs parts. Still others such as the Rochelle K-3385-EB, Champlain, Pompton, Placid, and Trocadero & San Raphael Power Lite Models, are even worse with complex and expensive proprietary parts often costing more than a new toilet would. Toilets such has the Pillow Talk, Palarre, San Raphael, Rochelle, Rialto, Cabernet, San Miguel, and Rosario, become more of a problem using proprietary fill valves, and flush valves making any thing more than a flapper replacement a headache. The flapper is more commonly available and even has aftermarket replacements leaving only the flush valve as a problem to get.
The Kohler One Piece Toilets such as the Gabrielle, Santa Rosa, and San Martine, have some models that aren’t too bad using a standard fill valve and a proprietary flush valve and flapper.
Even Kohler distributors will not have many of the parts in stock and ordering direct from Kohler is often the best option with a 3-week back order being the norm. Kohler has utilized so many proprietary parts that just getting parts can be a huge headache. Troubleshooting & Repairing Kohler One Piece Toilets is not the easiest task to do.