KCNA said the North was ready to start mass-producing the new medium-range ballistic missile, which it said is capable of reaching Japan and major American military bases.
The military said in a statement Tuesday that it has bolstered its air surveillance after the incident but provided few other details.
The United States, which has condemned repeated North Korean missile launches, said Sunday's launch of what North Korea dubbed the Pukguksong was of a "medium-range" missile, and US-based experts doubted the reliability of the relatively new solid-fuel type after so few tests.
The Security Council went on to emphasize the "vital importance" of North Korea's "immediately showing honest commitment to denuclearization" and demanded the DPRK conduct no further nuclear and ballistic missile tests.
Publicly-owned utilities will cut bills by £220 annually, says Labour
To those who register to vote in the June 8 election, they will be eligible to attend a secret party in London . I was elected by a very large number of members and supporters, ordinary people all over this country.
After a joint investigation with United States experts, Seoul said the recovered drones were flown from North Korea and had been pre-programmed to fly over South Korean military installations.
The United States has for weeks been negotiating with China, Pyongyang's ally, on a new sanctions resolution, but US Ambassador Nikki Haley said last week that no final draft text had been agreed.
The UN Security Council is due to meet on Tuesday behind closed doors to discuss Sunday's test, which defies Security Council resolutions and sanctions.
North Korea has in recent years touted its drone program, a relatively new addition to its arsenal.
The North has been also trying to develop nuclear missiles, which will be able to carry nuclear loads to the U.S. mainland.
Trump arrives in Bethlehem for Abbas meeting
The Palestinians want a state in the West Bank, Gaza Strip and east Jerusalem, captured by Israel a half century ago. Israel occupied east Jerusalem and the West Bank in 1967 in moves never recognised by the worldwide community.
The North's official KCNA news agency, citing the spokesman for the foreign ministry, said the country had "substantially displayed" the capabilities for mounting a nuclear attack on Hawaii and Alaska and had built full capabilities for attacking the US mainland.
Kim said "with pride" that the Pukguksong-2 was a "very accurate" missile and a "successful strategic weapon", KCNA said, adding he "approved the deployment of this weapon system for action".
Last week, North Korea repeated a vow to strengthen its nuclear strike capability as long as the U.S. maintains its "hostile policy" toward the country.
Solid-fuel missiles have their fuel loaded before being moved into place, allowing them to be launched faster and with more secrecy.
There was no immediate comment in North Korea's state-run media, and no reports that the North had returned fire.
Trump arrives for talks with Palestinian leader in Bethlehem
For Jews, the Western Wall, a retaining wall from the biblical Jewish Temple, is revered as the holiest site where Jews can pray. President Rivlin thanked Trump for his "commitment to Israel's ability to face the threats of today and of tomorrow".
So far nearly all the North's missiles have been liquid-fueled, meaning they must be time-consumingly filled with propellant before launch. But he said the US was in the early stages of applying economic and diplomatic pressure, and that perhaps North Korea was "acting out now in response to some of this pressure that I believe they're beginning to feel".