The North's drills involved a short-range missile launch but - unusually - the missile flew from a buried silo, which analysts say would help improve speed and stability in future tests of intercontinental ballistic missiles (ICBM).
State media KCNA said the exercises on Saturday and Sunday were designed to boost the country's "war deterrence and nuclear counterattack capability," accusing Washington and Seoul of making an "explicit attempt to unleash a war" against it.
"The drill also aimed to demonstrate our tougher will to make an actual war response and send a stronger warning to the enemy who expands their war drills for aggression," KCNA said.
In the exercises, a ballistic missile equipped with a mock nuclear warhead flew 800 km (497 miles) before hitting a target under the scenario of a tactical nuclear attack, KCNA said.
KCNA photos showed Kim attending the test, again with his young daughter, as flames roared from the soaring missile before it hit the target.
South Korea's defence ministry spokesman said the North is making significant technological advances in its nuclear programme but did not elaborate.
Kim said the exercises improved the military's war capability and urged the military to stand ready for any "immediate and overwhelming nuclear counterattack anytime."
"The present situation, in which the enemies are getting ever more pronounced in their moves for aggression against the DPRK, urgently requires the DPRK to bolster up its nuclear war deterrence exponentially," KCNA quoted him as saying.
"The nuclear force of the DPRK will strongly deter, control and manage the enemy's reckless moves and provocations with its high war readiness, and carry out its important mission without hesitation in case of any unwanted situation," he added, Reuters reported.
South Korea and Japan reported a launch of a North Korean short-range ballistic missile off the east coast on Sunday, the latest in a series of missile tests in recent weeks.
North Korea has reacted furiously to South Korea-US combined military drills, calling them a rehearsal for an invasion against it.
The allies have been carrying out exercises this month, including air and sea drills on Sunday involving U.S. B-1B bombers.
The US and South Korean navies and marine corps are set to kick off their first large-scale Ssangyong amphibious landing exercises in five years on Monday for a two-week run until April 3.
MNA/PR
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