I'm not an electrician, so maybe Jim can correct me if I'm wrong but...
You should really look into splitting the circuits in the Kitchen now or if/when you renovate the Kitchen. Kitchens use toasters, coffee makers, microwaves sometimes all at the same time that's too much for one circuit. You should have at least two GFCI protected circuits for counter top, alternating outlets to each. The dishwasher should be on it's own circuit. It sounds like someone may have added that dishwasher branch after the fact. If they chose wire without a ground it may be undersized as well. I have found ( and corrected) 14 wire spliced with wire nuts in the walls of 20amp circuits in my house. My house was built in 1968 and it has ground wires throughout. Jim, if that black flat wire (sounds like old romex) is indeed 12, can he run a ground to his copper plumbing?