His comments came hours after the launch which left analysts concluding that a wide swath of the United States, including Los Angeles and Chicago, is now in range of North Korean weapons.
The Korean Central News Agency said on Saturday that Kim expressed "great satisfaction" after the Hwasong-14 missile reached a maximum height of 3,725 kilometres and travelled 998 kilometres, before accurately landing in waters off Japan.
The agency quoted Kim as saying that the launch reaffirmed the reliability of the country's ICBM system and ability to fire at "random regions and locations at random times", with the "entire" US mainland now within range.
Analysts had estimated that the North's first ICBM on July 4 could have reached Alaska, and said that the latest missile appeared to extend that range significantly.
Immediately after the launch, US and South Korean forces conducted live-fire exercises.
Song Young-moo, South Korean defence minister, called for the deployment of strategic US military assets - which usually means stealth bombers and aircraft carriers - as well as additional launchers of an advanced US anti-missile system.
US President Donald Trump condemned the test as a threat to the world, and rejected North Korea's claim that nuclear power ensures its security.
"In reality, they have the opposite effect," he said in a statement.
Weapons and tests "further isolate North Korea, weaken its economy, and deprive its people"," he said, as he vowed to "take all necessary steps" to ensure the security of the US and its allies.
China meanwhile, North Korea's most important ally, urged Pyongyang to abide by the relevant UN Security Council resolutions and refrain from escalating tensions, according to a foreign ministry spokesman quoted by the Xinhua news agency.
Washington and its allies have watched with growing concern as Pyongyang has progressed towards its goal of having all of the US within range of its missiles, to counter what it labels as US aggression.
There are other hurdles, including building nuclear warheads to fit on those missiles and ensuring reliability.
But many analysts have been surprised by how quickly Kim has developed North Korea's nuclear and missile programmes, despite several rounds of United Nations Security Council sanctions that have squeezed the impoverished country's economy.
North Korea created a stir on July 4 when it test-fired its first ICBM, a Hwasong-14 missile, which experts believe could have the potential to reach Alaska.
Kim, who personally oversaw that launch on America's Independence Day, described it as a gift to the "American bastards".
North Korea is not believed to have yet developed the technology to miniaturise a nuclear weapon to fit in a missile's warhead.
Source: AP news agency