This is a question I've been much concerned about. I've read a bit of Korean history and such, but nothing compared to the article provided by the following link:
"..... It's unlikely South Korea would ask the U.S. to launch a nuclear counter-strike. It would be an all out conventional version consisting of missiles launched by SSN's, Cruisers along with naval and air bombers from the U.S. and other allies. Even if there were multiple nuclear warhead detonations in the south, doing so in the north would mean there would be entire provinces all over the peninsula that would be uninhabitable. North Korean nuclear warhead technology is not considered as advanced as western technology. It's estimated that the largest nuclear device the North Koreans have built is a warhead with an approximate blast yield of 5 Kilotons, roughly 1/4 the destructive yield of a World War II era Mark III (Fat Boy - used to bomb Nagasaki, 21 kt) . There would be no mercy shown as radiation poisoning would last 50+ years. South Korea would ask the U.S. to target Kim Jong-un and every government installation known in addition to military targets. And it would not end until the leadership was wiped out. China and Russia would probably not allow Kim Jong-un asylum and in fact, may arrest him if he attempted it and allow him to be put on trial for war crimes......"