Having just replaced the diverter on my K-10412, and (finally!) getting the faucet to work correctly, let me add a couple of comments:
Check out the parts diagram at Kohler Plumbing
Make sure the old diverter is intact upon removal. If not, well... those little parts have to go somewhere; so there's a good chance the missing pieces are stuck in the line. While several culprits could cause a clogged line, the diverter pieces are prime suspects. Make sure all the parts are retrieved before inserting the new diverter and re-assembling the faucet. If the little rubber gasket from the diverter is lodged in the water line, neither a water flush nor compressed air will remove it. (At least it didn't move mine.) I had to disconnect the hose from the bottom of the faucet and snag the gasket with a wire and pull it out.
Also, here's the sequence for disassembly:
Place your drain covers or strainers over the drains. There are a couple of little parts that can fall down the drains. ( I learned this the "hard way" too.)
1. Handle set-screw cover (It's plastic, it just snaps in. Pry it out with a knife, probably best from the top or bottom as there are little securing tabs on either side. And yes, there is a "top" and "bottom" to it. That is to say, when you put it back upside down, it won't fit correctly. )
2. Handle (Loosen set screw with allen wrench.)
3. Bonnet (Should be threaded only hand tight. Wrap some tape on your plier jaws if you need to break it loose.)
4. Threaded ring (This is the big one that will accept a wrench.)
5. Valve (White cylinder about an inch tall. It just lifts out.)
At this point, you should be looking down into the brass body, which has three holes in it.
6. Spout. (The spout should lift straight up off the body. There is nothing holding it in place. If you must, pry gently under to spout. There's a nylon washer there, try not to damage it.)
7. Diverter (Inserted into the back of the body. Pliers may be necessary to get the old one out.)
Make sure you have all the pieces to the old diverter when you take it out. If not, start looking. They're in there... somewhere.
Insert the new diverter and reassemble in reverse order. You'll be done before dinner :-)
Good luck, and thanks to those who have contributed previously.