Single Handle faucet broke on my Kohler kitchen faucet and can find no replacement or solution

Is cheaper to replace the whole tap as the service kits cost almost as much as the tap to buy, I presume you are talking about a flick mixer tap (that is what they are called in australia)
Replacing them is reasonably easy, turn off water at the main out on the footpath turn the tap on to let the water run out, go under the sink with a couple of spanners or deep reach socket and undo the lines and the tap hold down nuts and remove, replace in reverse order