First things first: I was wrong about Kim having no sons; he has three (one with his estranged wife, two with a now-dead mistress).
A few readers asked questions about succession--was anyone in line to take the mantle?
It seems Kim's oldest son fell out of favor when he was caught with a fake passport in Japan in 2001, and some rumors have flown around that his second son is being groomed for the position. But they have not made public appearances with their father (unlike Kim Jong Il with his father in the late 1980's), and have remained largely in the background. At this point, with Kim probably in the hospital recovering from a stroke, North Korea's party elite may have already chosen a leader to succeed Kim when he dies.
Professors on the subject debate some between a factional internal struggle for power, and a collective ruling effort on the part of the military (which could well be one very slow and messy factional struggle for power). They agree that Kim's sons are unlikely to take the lead themselves, but may remain as a sort of British/Chinese/Japanese symbolic monarchy for the purposes of morale and legitimacy to the military rulers.
But really, little is known. Most analysts try to decipher what they can based on short clips of public footage--who is standing closest to Kim, who is talking to him, etc. South Korean intelligence agencies don't predict an imminent collapse, but the South Korean executive branch held an emergency meeting to discuss the possibility of refugees.
The worst-case scenario I could imagine is the following: the North Korean military is the fifth-largest in the world, and is very loyal to Kim. If Kim has failed to hedge their loyalty to a second person, then military regulars may default their loyalty to elite generals. If there is disagreement over the particular elite general that should take the helm, a warlord-style civil war is not impossible, and would likely lead to North Korean collapse, massive refugee flows, and Chinese occupation. The best-case scenario is probably that a reformist or realpolitik ruler is able to slide into power quite easily--someone more stable and rational than Kim would mean relaxed relations with the US, China, and South Korea, and potentially improved trade relations that could modernize the country.