Kim expressed "satisfaction" over the weekend drills, which were held to "let relevant units get familiar with the procedures and processes for implementing their tactical nuclear attack missions", the report said.
The drills were the fourth show of force from Pyongyang in a week and came as South Korea and the United States stage their own military manoeuvres -- 11 days of joint drills known as Freedom Shield, their largest in five years.
North Korea views all such exercises as rehearsals for invasion and has repeatedly warned it would take "overwhelming" action in response.
Saturday and Sunday's drills were divided into exercises simulating the shift to a nuclear counterattack posture and a drill for "launching a tactical ballistic missile tipped with a mock nuclear warhead", KCNA said.
"The missile was tipped with a test warhead simulating a nuclear warhead," it added later, without giving further details.
Seoul's joint chiefs of staff said Sunday that the short-range ballistic missile fired by Pyongyang flew 800 kilometres (500 miles) before landing in the East Sea, also known as the Sea of Japan.
They branded it a "serious provocation" that violated United Nations sanctions and said it was being analysed by US and South Korean intelligence.
Tokyo has also condemned the launch, as did the US military's Indo-Pacific Command.
Seoul and Washington have ramped up defence cooperation in the face of growing military and nuclear threats from the North, which has conducted a series of banned weapons tests in recent months.
The flurry of aggression by Pyongyang has also pushed Seoul and Tokyo to mend fences over historical disputes and try to boost security cooperation.
On Thursday, Pyongyang test-fired its largest and most powerful intercontinental ballistic missile, the Hwasong-17, its second ICBM test this year.
The ICBM launch followed two short-range ballistic missiles on Tuesday and two strategic cruise missiles fired from a submarine last Sunday.
The UN Security Council is expected to hold an emergency meeting Monday over the ICBM launch at the request of the United States and Japan, Yonhap news agency reported.
Kim said the drills on Saturday and Sunday had filled the North Korean military units "with great confidence", according to KCNA.
He also noted that the North "cannot actually deter a war with the mere fact that it is a nuclear weapons state," and said that the country could only reach its goals "when the nuclear force is perfected as a means of actually capable of mounting an attack on the enemy."
North Korea declared itself last year an "irreversible" nuclear power and Kim recently called for an exponential increase in weapons production, including tactical nuclear weapons.
Kim also ordered the North Korean military this month to intensify drills to prepare for a "real war".
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