They say no single person in the communist dictatorship is poised to take over as smoothly as Kim - groomed for 20 years - did after his father died in 1994. But the 62-year-old Jang, husband of Kim's younger sister, is seen as a potentially critical player.
Jang heads the administrative department of the all-powerful Workers' Party. More importantly, he oversees the intelligence agency and other military-related institutions, analysts say.
"Jang could play a key role in a collective leadership because he's someone who would be able to bond the military and the party together," said Hong Hyun-ik of the Sejong Research Institute, South Korea's security think tank.
A technocrat educated in Russia during Soviet times, Jang was a rising star until he was summarily demoted in early 2004 in what analysts believe was a warning from Kim against gathering too much influence.
But Kim got Jang back at his side in 2006.
"In such a power structure, the most reliable person is a relative," said North Korea expert Koh Yu-hwan of Seoul's Dongguk University. "He is now No. 2."
Jang's two brothers served in high-level military posts, which could give him connections to the military, Koh said.
Other senior figures in a collective leadership would likely include Defense Minister Kim Il Chol and other top military and party figures, the analysts said.
But with no way of confirming the speculation, all the guesses could be wrong, Koh cautioned.
"Who knows?" he said. "Somebody we don't know may actually be really powerful behind the scene."
The analysts also warn against assuming for a fact that 66-year-old Kim is incapacitated or dying.
South Korean and U.S. officials say their intelligence sources confirm Kim suffered a stroke in August and underwent surgery. Last week, the chief of Seoul's main spy agency told lawmakers that Kim's condition appeared to have "improved a little."
North Korean officials never talk about their leader's health, except to strenuously deny he was felled by a stroke.
The reports have led some to see a link between Kim's health and his country's push to restart its nuclear program - restoring equipment, testing its reactor engine, ordering U.N. inspectors to remove monitoring equipment and keep out of the complex.
The regime's abandonment of the painstakingly crafted disarmament-for-aid deal after steadily disabling the program since last November has raised speculation that if Kim is ill, the hard-line military is quietly pulling the strings.
But others say the recent escalation of tensions is characteristic of Kim's strategy.
"Things are moving this way precisely because Kim Jong Il is making the policy decisions," said Hong, at the Sejong institute.
He called it Kim's "typical brinkmanship strategy" during diplomatic talks.
Analysts say North Korea's recent moves are no surprise because the regime has established a pattern of raising tensions when negotiations reach a deadlock and its demands aren't met.
North Korea says it abandoned the February 2007 disarmament deal last month - just weeks after blowing up a cooling tower in a dramatic show of its commitment to the pact - because Washington failed to deliver on a promise to remove the regime from the U.S.
"I think North Korea is likely to keep a hard-line stance on the United States for the time being," said Dongguk University's Kim Yong-hyun. "It's good for solidifying internal unity amid speculation about Kim's illness."