How do I remove this stuck faucet valve cartridge?
My cold side cartridge needed replacement. (My valve assembly has manufacture date of 2007.) I have a number of same Kohler Lavatory sets and have had to replace a few valve assemblies a few times. Kohler has sent me entire retrofit valve set because many times, as was described in prior comments, it's difficult to break the seal of the top nut to get at and replace the cartridge. I decided to get brave after reading the ideas from comments and decided to try to get at the cartridge. 

This is what worked for me. I used a vice grip and attached it right below the threads of the assembly (just above where the water line attaches to the assembly). It's machined there to accept a wrench and the vice grip secures perfectly. I took my ball pain hammer and tapped the nut area all around to hopefully break the seal. I attached a 5/8" socket and ratchet wrench to the cartridge top nut (You do not have to take off that cotter pin). I then placed the valve assembly on the garage floor with the vice grip attached. I then secured the 5/8" socket to the top nut and used one foot to make sure the valve set doesn't move and used my other foot to apply leverage to the socket wrench and with a few stomps on the ratchet handle the nut loosened and the cartridge came out of the assembly.