Wednesday's missile test near the east coast, aimed at assessing the country's strategic technology for developing a new type of high-altitude missile, destroyed targets in the air from 200km away, KCNA said.
South Korea's Joint Chiefs of Staff confirmed the missile test occurred at 5pm (0800 GMT), Yonhap News Agency reported.
The South Korean military had been aware of launch preparations and was ready for it, Yonhap said, citing a Joint Chiefs official.
Kim observed construction at another site on an 8700-ton nuclear-powered submarine capable of launching surface-to-air missiles, KCNA said. It did not identify the location or date of his visit.
The submarine project is part of the North Korean ruling party's effort to modernise the country's navy, one of five key policies the party is pushing to develop its defence capabilities, KCNA said.
A KCNA photo showed Kim at a submarine construction site with his daughter, Ju Ae.
Surrounded by other officials, he smiles as the teenager, considered by some analysts as the frontrunner to succeed her father, stands next to him in an indoor facility housing a red vessel.
Kim said North Korea is building multiple attack destroyers and nuclear submarines and working to rapidly accelerate construction so that vessels can be equipped with various weapons, according to KCNA.
The design of the submarine's hull indicates it has been equipped with a nuclear reactor, and the vessel is almost ready to sail, Hong Min, a senior research fellow at the Seoul-based Korea Institute for National Unification, wrote in a report on Thursday.
Kim was quoted as saying the all-out development of nuclear capabilities and modernisation of the navy were essential and inevitable, while "the present world is by no means peaceful".
He said South Korea's plan for developing a nuclear submarine, agreed with Washington, would further inflame tensions on the Korean Peninsula and poses a risk to national security that requires him to take action.
North Korean state media also criticised the recent entry of a US nuclear-powered submarine into a South Korean port, calling it "an act of escalating military tensions" on the Korean Peninsula and in the region.
On Tuesday, the nuclear-powered submarine USS Greeneville arrived at the Busan port for crew shore leave and the loading of supplies, the South Korean Navy said.
North Korea also said earlier this week that Japan was showing its intention to possess nuclear weapons, encouraged by South Korea's moves to develop a nuclear submarine.
