I think a better project would be to remove the entire countertop and replace with something that is solid surface. You'll use the existing countertop as a template for the solid surface replacement. When you are done you'll have something that looks great, is impervious to water and you can remount the undermount sink and use this as an occasion to replace what is probably a less than stellar kitchen faucet with something nicer looking and with better utility. It will be a bit of work, but you probably won't deal with this again.