The Ariston GL4 is a 4 gallon heater. It should provide enough for the sink but I don't know how many gallons the dishwasher uses. If the dishwasher has a heater (most do) that will make up for the fact that some of the water will be warm. It will probably extend the cycle.
You should be able to re-route a hot water line using PEX without tearing up too much. I assume that you are in a moderate climate becuase most houses where it is really cold have basements. By running a hot water line to the POU heater, you get instant hot water even with long runs, and the line makes up with hot when it finally gets there.
You should find the shortest route, probably through the attic and maybe from a nearby bathroom.
If your existing heater is gas, you will want to use that because of lower fuel charges. If your existing water heater is electric, you might want to just put a 10 gallon heater in the attic or some place near the kitchen, instead of the Ariston, and abandon the hot water line to the kitchen.
If you have one leak, you should consider that the first of others likely to come. You might want to hold off the re-route until you get another, or plan to re-route both hot and cold now.