Pyongyang said Sunday's test was made all the more necessary by "recent geopolitical crisis" -- a clear nod to this weekend's events in North Korea's socialist ally Venezuela.
Seoul and Tokyo said Sunday they had detected the launch from near Pyongyang of two ballistic missiles.
It was the country's first test of the year and came just hours before South Korean leader Lee Jae Myung headed to Beijing for a summit.
Lee has said he hopes to use the visit to harness China's clout over North Korea to improve Seoul's ties with Pyongyang.
On Monday, the Korean Central News Agency (KCNA) quoted Kim as saying the test showed "the readiness of the DPRK's nuclear forces", referring to North Korea by its official name.
"Important achievements have been recently made in putting our nuclear forces on a practical basis and preparing them for an actual war," Kim was quoted as saying by KCNA.
Pyongyang's activity, he said, was "aimed at gradually putting the nuclear war deterrent on a high-developed basis".
"Why it is necessary is exemplified by the recent geopolitical crisis and complicated international events," he said, a clear reference to Washington's weekend capture of Venezuelan President Nicolas Maduro.
The operation represents a nightmare scenario for North Korea's leadership, which has long feared a so-called "decapitation strike" of that kind and accused Washington of seeking to remove it from power.
Pyongyang has for decades justified its nuclear and missile programmes as a deterrent against alleged regime change efforts by Washington.
North Korea's new weapon system using hypersonic missiles was first tested in October.
Hypersonic missiles travel at more than five times the speed of sound and can manoeuvre mid-flight, making them harder to track and intercept.
They have been deployed to deadly effect this year on cities in Ukraine by Russia, with whom North Korea has deepened ties in recent years, and by Iran against Israel.
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