Turn off the two water-supply valves under the kitchen sink. Turn the valves' handles clockwise to close them.
Lift up on the Kohler K-10412's handle to allow all remaining water in the unit to trickle out of the spout and relieve the water pressure in the faucet.
Pry the plastic cap from the rear of the handle with a slotted screwdriver to reveal the handle's set screw.
Loosen the set screw with an 1/8-inch hex wrench. Turn the screw two full revolutions counterclockwise. Pull the handle off the faucet.
Lift the bonnet from the faucet; if the bonnet sticks, carefully pry it up with the slotted screwdriver.
Loosen the mounting nut holding the K-10412's cartridge valve in the faucet body with an adjustable wrench. Turn the adjustable wrench counterclockwise to loosen the mounting nut. Remove the mounting nut.
Lift the K-10412's cartridge valve and spout off the faucet body by tugging straight up gently.
Lift the old diverter out of the faucet body with needle-nose pliers.
Prepare the K-10412's new diverter for installation. Push the O-rings onto each end of the new diverter, Kohler #75878. Apply a small amount of plumber's grease to each O-ring.
Place the new diverter in the slot in the bottom of the faucet body so the fat end of the diverter faces the rear of the faucet body.
Inspect the O-ring on the faucet body for damage. Replace the O-ring and lubricate the O-ring with plumber's grease, if needed.
Slide the spout over the K-10412's faucet body so it faces the sink's basin.
Lubricate the cartridge valve with plumber's grease and slide the valve into the faucet body so the valve's handle adapter faces the front of the faucet body.
Slip the K-10412's mounting nut over the cartridge valve and thread the nut into the faucet body. Tighten the mounting nut with the adjustable wrench.
Slide the K-10412's bonnet over the mounting nut. Push the bonnet onto the faucet body.
Push the faucet's handle onto the cartridge valve's handle adapter so the handle faces the front of the faucet.
Tighten the handle's set screw with the hex wrench and replace the cap.