The tests were conducted on Friday as part of an assessment of the newly built warship's combat systems, state media KCNA reported on Sunday.
They included checks of target-detection and information-processing capabilities, integrated firepower systems, naval guns, automatic cannons and electronic warfare equipment, KCNA said.
Kim was briefed on the vessel's weapons evaluation program before watching the strategic cruise missile launch and other tests, according to the report.
Kim hailed recent advances in weapons development and called for further efforts to expand North Korea's war deterrence and combat capabilities, KCNA said.
He ordered officials to complete trials of the Kang Kon and commission the destroyer into naval service within two months, KCNA said.
In June North Korea commissioned the 5000-tonne destroyer Choe Hyon and announced plans to deploy the Kang Kon, a vessel of the same class, as part of a broader naval modernisation drive.
After years of prioritising ballistic missile development, Kim has increasingly turned his focus to naval capabilities, including the construction of a nuclear-powered submarine. While outlining his five-year military goals at February's Workers' Party congress, Kim also called for intercontinental ballistic missiles capable of underwater launches.
Kim has called for the construction of two warships of the class every year over the next five years and also outlined plans for larger 10,000-tonne warships.
The Kang Kon was repaired last year after partially capsizing during a launch ceremony.
North Korea has increasingly highlighted naval development as it seeks to bolster what Kim has described as the weakest branch of its armed forces.
with AP
