The Korean Central News Agency also published photos indicating that it is developing a new ballistic missile.
Prior to Trump's speech, U.S. Secretary of State Rex Tillerson suggested in Washington the same day that dialogue with North Korea may be possible in the near future, as he hailed the restraint Pyongyang has shown recently with its weapons programs.
Under Kim, Pyongyang has made rapid strides in its ballistic missile technology, which it is banned from pursuing under United Nations resolutions that have slapped it with seven sets of sanctions.
A series of threats followed from both sides, and while the rhetoric has since eased, the US and South Korea this week kicked off their annual Ulchi Freedom Guardian military drills, which the North always condemns as dress rehearsals for invasion.
North Korean leader Kim Jong Un reportedly came out of hiding early this month to make a secret visit to a military unit near the rogue state's border with the South.
However later North Korea's state media reported that its leader had chose to wait and see what the United States does next.
The institute produces intercontinental ballistic missile warheads and also develops carbon compound materials used in rocket engines.
However, the report does not mention when Kim visited the institute where he was briefed about the process of manufacturing of intercontinental ballistic missile (ICBM) warhead tips and rocket engines.
Solid-fuel engines help cut the required preparation time for a missile launch and make it more hard for outside observers to detect a launch in advance.
Carbon-carbon composites are used to manufacture missile warhead tips and engine exhaust nozzles, KCNA said.
President Donald Trump reflected on his recent verbal jousts with North Korea during a campaign rally in Arizona on Tuesday, claiming that the country's bellicose leader was "starting to respect" the United States.
The North has threatened to fire a salvo of missiles toward the US Pacific territory of Guam, but has since backed away from the plan.
One showed a missile called Pukguksong-3, which appears to be the latest in the Pukguksong, or Polaris, series.
North Korea has conducted two nuclear tests and dozens of missile tests since the start of previous year, significantly raising tensions on the heavily militarized Korean peninsula.