The South Koreans must be shitting themselves right now.
The line of succession isn't all that clear. Yes, he may have favored Kim Jong Un, but he's also his 3rd son. And the other two are still in North Korea.
The North Korean people believe that Kim Jong Il and Kim Il Sung were demigods (complete with magic powers - that extended beyond the grave, in the case of Il Sung). They also firmly believe that the South Koreans (and the U.S.) are to blame for all their economic troubles, famines, etc. Every time Kim Jong Il had so much as a cold, North Korean TV would report that a South Korean/U.S. assassination plot was the cause, but that the Dear Leader, through sheer willpower, had survived.
And let's not forget they have nukes. Their crude missiles may not be able to go the 800 or so miles to Tokyo. But I guarantee they can make the 100 miles to Seoul.
Whatever happens up there, it's bound to spill over the South Korean border. Whether it's like Berlin circa 1989 or Cambodia in the 1970's.