Our kitchen sink is a Delta Ashton 19922-SSSD-DST we know from our blog post for when it was installed. Naturally it is no longer available, but the cartridge that controls the flow of water is available: Delta RP50587 Single Handle Valve Cartridge, from several sources, including Home Depot.
To replace the cartridge, we first remove the handle. There is a little button with blue and red to indicate cold and hot and we can pry it up. Under it is a 1/8 inch hex set screw.
The dome like stainless steel covering just unscrews -- it's just decorative.
At this point, be sure you have turned off the water supply, both hot and cold, to the faucet. The next step removes the cartridge. You can turn the water off at any time before this.
Once the dome is removed, it exposes a big copper hex nut. Using a big wrench, remove that.
Now the old cartridge just pops right out. Replace it with the new cartridge, and reverse the steps -- big copper hex nut, then the dome, then the handle, and tighten the set screw, and replace the little button. Turn on the water and test it out.
Looks just like it did before, and with luck, it will be another 9 years before we have to do this again.