The Don has a handful of kinetic options to address the Little Rocket Man, but surely the Supreme Leader has a few tricks up his sleeve as well. These options would be evaluated within the context of escalation control [don't want things to get out of hand] and escalation dominance [finding the option that would be the last move, and a winning move].
Both escalation dominance and escalation control take place within the context of bounded rationality [neither side has perfect knowledge of their adversary's intentions or capabilities, and both sides have value preferences that shape their worldviews]. Both sides are bureaucratic actors, who must satisfice a variety of domestic and international stakeholders. Neither side is a perfectly "rational" actor who can optimize outcomes for a single variable. Indeed, both sides are incentivized to proclaim their irrationality, and both sides were doing a pretty good job of this in the warmup rounds.
An all-out war on the Korean peninsula would result in the prompt collapse of the Korean People's Army [within a few days] and the implosion of the North Korean regime. The North might initiate the use of nuclear weapons to signal to China and Russia that they urgently needed to prop up the regime [as South Africa imagined using nuclear weapons to signal to other white countries that it needed a bulwark against African Communists].
The United States might retaliate with nuclear strikes against North Korea. The DPRK is a target-poor environment, with relatively few aimpoints that fall within traditional categories of counter-force, counter-value, counter-recovery, and so forth. In the Cold War, the Americans would have been content to leave the Soviet Union a "smoking, radiating ruin". But Japan is down-wind of the DPRK, and the South has plans for the North that do not encompass inheriting the world's largest burn hospital.
Probably no one wants to go there, but both sides are contemplating the middling rungs that are kinetic but not hyper-kinetic. The American's have a few not-implausible options, as do the North Koreans. Many are familiar, while others might ratchet things up a bit.
1 - The regime a long history of planned assassinations and commando style attacks. In 1968, a team of 31 North Korean commandos were sent to the South to infiltrate the Blue House (the ROK presidential residence) and assassinate President Park Chung-hee. Kim Jong-nam, the half-brother of North Korean leader Kim Jong-un, was killed in Malaysia in February 2017. North Korean defectors have also been targets of assassination attempts.
2 - The North has already demonstrated a capacity and a willingness to engage in a variety of deadly provocations. A North Korean torpedo sank the Cheonan, a South Korean navy ship, in March 2010 with the loss of 46 sailors. In November 2010, South Korea returned fire after North Korea fired dozens of artillery shells at Yeonpyeong island, one the South's border islands, killing two marines.
3 - Korean Airlines plane flight YS-11 was hijacked and flown to North Korea in 1969. North Korea freed 39 passengers on the plane, but 11 crew members and passengers were not returned. On 29 November 1987, North Korean agents hid a bomb on Korean Airlines Flight 858, killing all 115 on board.
4 - Four North Korean fighter jets intercepted a U.S. Air Force plane in international airspace over the Sea of Japan early 02 March 2003, and similar encounters followed. Nothing came of it then, but the DPRK has an air force with fighters, and long range anti-aircraft missiles. Both could be used to make airspace adjacent to the North hazardous to aerial navigation.
North Korea does not have a declared Air Defense Identification Zone [ADIZ], and when China declared an ADIZ in the East China Sea on November 23, 2013, the zone overlapped with the pre-existing Japanese ADIZ. There are no internationally agreed rules of engagment for an ADIZ, but the standard procedure is that commercial aircraft identify themselves in advance, military aircraft do not, any air are subject to being intercepted and inspected, but all are allowed innocent passage. The North could do all kinds of annoying things with an ADIZ, as Muammar Gaddafi demonstrated when he proclaimed the Gulf of Sidra as its territorial waters and subsequently declared a "line of death". The USA crossed that line from 1981 and 1987, and various skirmishes resulted and aircraft downed and buildings bombed.
5 - South Korea is stuck between the North and the Americans, literally and politically. North Korea [and Russia and China] would like to eject the Americans from the Peniinsula, and would like to drive wedges between the ROK and USA. Any kinetic actions that created daylight between the USA and ROK would be to the good, from the regime's perspective. Kinetic actions aimed at the Americans, accompanied by kissy-face engagement with the South, might have some attraction. A limited missile attack on an American military based in the South, accompanied by proposing a summit between the leaders of the North and South, might be an attractive option.
6 - Restricted Submarine Warfare - Further up the escalation ladder, the North might declare a limited quarantine of the South, excluding American shipping. The South has dozens of diesel submarines, and could probably give the American blockade runners a run for their money. For American military vessels in waters immediately adjacent to the South, this might provide valuable training opportunities. But commercial shipping in Indo-Pacific sealanes would be another matter. The North would be challenged to identify an "American" freighter, given flags of convenience, but surely some container ship could be singled out for attention, and the maritime trade with the South greatly impacted.
7 - The North already has a demonstrated capacity and willingness to use offensive information operations, witness their attack on the movie "The Interview". The DPRK cannot make sewers flow backwards or other such cartoonish "Digital Pearl Harbor" exploits, but surely they can be annoying to American interests.
8 - The North announced an intent to conduct long-range missile tests that would bracket the American territory of Guam. Having a brace of North Korean reentry vehicles splashing down all in water adjacent to Guam would surely get people's attention. Some advance Notice to Aviators and Mariners [NOTAM] would make it street legal, and make sure that the audience was lined up for the show.
9 - Jumping over a number of rungs in the escalation ladder, the North could conduct an above ground nuclear test. It has been some years since there have been nuclear detonations other than underground tests, and such a test, with a grinning Supreme Leader in the foreground, would surely cause consternation.
10 - The North could credibly threaten to nuke an American city, to provoke a spontaneous evacuation of the populace. Make the threat a few days in advance, and then jerk people around by changing the target city and the date of attack.
11 - Further vexation might result from a nuclear test detonated at high altitude over Guam. A low yield detonation [a few kilotons] at high altitude [tens of kilometers] would get everyone's attention, but probably not produce appreciable physical damage to Guam or the citizens thereof. Some thought might be required to anticipate flash blindness or Electromagnetic Pulse [EMP] effects, but these might not be particularly severe. The Americans might be at a loss to respond.