All you need for it is an extra deep socket which I have. I got the cold water one out with no problem. This one seems to be rusted in or stuck or something like that.They make a cartridge removal tool that you can purchase at your local plumbing store. Check with a neighbor who is handy at plumbing, maybe he might have one.
There is no nut for this, because the cartridge itself is threaded and screws into the base. So all you need is a socket wrench because once it's unscrewed the entire cartridge is coming out. I've been going as Rambo on it as I can and apart from yanking the entire sink away from the counter top I've got nothing.The cartridge removal tool isnt for taking the nut off. It screws into the stem of the cartridge and assists in pulling it out. Sometimes cartridges are a real PIA to remove. A Moen rep told me to spray hairspray on their cartridges if they are stuck. Apparently the hairspray shrinks the rubber seals enough to get them to break free. Perhaps this will work with Kohler...not sure. Sometimes you just have to get "Rambo-ish" with them and yank them like a tooth with a pair of channel locks.