Try Googling (kohler)(K-5810-4-7)(manual) without parens.
How do I replace a cartridge for kohler bancroft sink faucet?
kohler bancroft is the sink manufacturer, not the faucet mfgr. do you have a clue what faucet mfgr it might be?
I had cascade toilets with a pull handle. Once I unscrewed the handle, there was a nut that held the lid on. Remove that, then the lid, then you should be able to get to the mechanism.
Check the inside of the faucet for debris blocking the cartridge or the inlets.
Have a 1256315 Kohler Touchless Flush toilet. Inserted new batteries and now it does not flush. Checked everything, wire is not bent or jammed and batteries are inserted correctly. Help!!!
If it is a pull spout there is a screen and or check valve assembly where the hose connects.
Call Kohler for a new faucet.
You can cut a new seat first which is good in most cases . I would not do what you are saying you are going to make your self problems . Reseat if it don't work get a new tap.
Look on line. There are whole bunch information there.
Read and understand before attempting. Make sure water is off, wife will not be happy.
Call Kohler if you need more info.