State media said Kim supervised the launch of Pyongyang's first intercontinental ballistic missile (ICBM) Tuesday, which it says is powerful enough to reach the US mainland.
"With a broad smile on his face," Kim called on officials to "frequently send big and small 'gift packages' to the Yankees," KCNA reported, as it listed the technical successes of the rocket, identified by the North Koreans as a Hwasong-14.
The report said the missile was able to carry a "large-sized heavy nuclear warhead," and despite "extreme overload and vibration the nuclear warhead detonation control device successfully worked."
Among the elements being tested was a warhead tip "made of newly developed domestic carbon compound material" designed to withstand the extreme heat of re-entry to the Earth's atmosphere.
It added "the warhead accurately hit the targeted waters without any structural breakdown at the end of its flight."
North Korea said the missile flew on a steep trajectory, going 2,800 kilometers (1,741 miles) above the Earth, before splashing down in sea off the Korean Peninsula 930 kilometers (578 miles) from its launch site.
The missile was launched Tuesday from Panghyon, in North Pyongan province, and landed in the sea off the Korean Peninsula.
How true are the claims?
Claims of unbridled success by North Korea's state media need to be taken with a pinch of salt, said Melissa Hanham, a senior research associate at the James Martin Center for Nonproliferation Studies.
"We can never take KCNA exclusively as a source because its so prone to embellishment, (and) the information it reports can never be confirmed.
"On the other hand, it's not out of the realm of possibility... I think its best to assume that they have successfully tested an ICBM. The reason they're sharing this technical data (through state media) is to prove that they have it," she said.
The Pentagon late Tuesday confirmed North Korea's test was of an ICBM
, and South Korea's intelligence services Wednesday confirmed the suspicion that its hostile neighbor fired a missile with a range greater than 5,500 kilometers -- the generally accepted lower limit of an intercontinental ballistic missile.
Range growing by each test
Hanham said analysts are now examining the images provided by North Korean state media, to look for similarities to previously launched missiles.
The Hwasong-14 tested on July 4 is similar to the Hwasong-12, which was test-fired in May, but perhaps with a larger engine configuration and an extra stage, a section of the missile that's released during flight. "The second stage looks like something we haven't seen before," she said.
While the test seems to have indicated the missile's range was at least 6,000 kilometers, Hanham said its maximum potential could be even further. "The scary thing is, (we don't know if) they even tested it to its full range," she said.
Another class of missile, the KN-08, aka the Hwasong-13, which was displayed during a North Korean military parade in April could have even greater potential.
Based on analysis of its visible fuel and oxidizer tanks, it could hit as far away as Washington DC, though that class of missile is yet to be tested, she said.
David Wright, co-director and senior scientist at the global security program with the Union of Concerned Scientists (UCS), said North Korean rocket scientists seem to be making advances on multiple fronts.
"One of the things that's interesting, watching from a technical point of view, is that it has eight or nine different missiles in development in parallel," he told CNN.
"They're making progress and they have a lot of things in the workshop that they're putting together."
Warhead that can fit on an ICBM
There is a growing consensus among analysts that North Korea has the ability to build a warhead that can fit onto a missile.
Five nuclear tests over the past 11 years suggest that the regime has indeed developed nuclear weaponry, and many analysts now believe that the miniaturization process is progressing rapidly.
If images released by KCNA in March 2016, showing Kim posing with what appeared to be a nuclear warhead, are to be believed, progress has indeed been made on this front.
The UCS' Wright thinks that while they may not yet have succeeded in producing a warhead capable of being attached to their new class of ICBM, the clock is ticking.
"The big question is whether or not they can build something that's both small enough and rugged enough to withstand the flight of a long range missile," Wright said.
"That could be a year or so. It's hard to tell. But it's clear that unless something changes that they're on their way to both a long range missile and a warhead to put on it.
"And I would argue that that's exactly why the United States needs to be finding a way to talk to North Korea to basically put a cap on this program."