The North has expressed frustration that the South won't defy international sanctions and resume South Korean tours at the site.
The statement comes after North Korea earlier this month broke off working-level talks with the United States in Sweden, pushing denuclearisation negotiations back into limbo after what had been months of stalemate.
Mt Kumgang was one of two major inter-Korean economic projects, along with the Kaesong industrial zone, and an important token of cooperation between the two Koreas during decades of hostilities following the Korean War.
North Korea has also ignored South Korean calls for dialogue recently and is seen as trying to force Seoul to make stronger efforts to coax major concessions from Washington on its behalf.
Trump described the letter as "great" and said it was hand-delivered to him on August 8.
Pyongyang had warned that the war games would jeopardise nuclear negotiations between the United States and North Korea.
Kim personally organised and guided the firing of the "state-of-the-art weaponry system" on July 25 and was "gratified" with the outcome, KCNA said.
The two leaders are said to have met in a surprise meeting at the Demilitarised Zone, that separates North and South Korea
Trump became the first sitting US President to step on North Korean soil.
US-North Korea talks have been stalled since a failed summit in February between Trump and the North Korean leader in Hanoi, Vietnam.
Choe Son Hui said North Korea's resolve for denuclearisation remained unchanged but it will only happen if the United States changes its current calculations.
The first face-to-face talks between Putin and Kim, held on an island off the Russian Pacific city of Vladivostok on Thursday, did not appear to have yielded any major breakthrough.
The young leader will meet with Putin in the Russian Pacific port of Vladivostok on April 25 with the nuclear stalemate with the United States topping the agenda.
The announcement comes after North Korea announced it had tested a new tactical weapon with a "powerful warhead", as denuclearisation talks with Washington appear to stall.
Trump and Kim have met twice, in Hanoi in February and Singapore in June, building goodwill but failing to agree on a deal to lift sanctions in exchange for North Korea abandoning its nuclear and missile programs.
In an expected move, Kim Jong Un was re-elected as chairman of the State Affairs Commission at a session of North Korea's rubber-stamp legislature that took place on Thursday, Korean Central News Agency (KCNA) said.
It was the first time Kim stated North Korea's position on the second U.S.-North Korea summit in Hanoi that collapsed in February, and signalled a continued focus on economic development, a strategic direction officially declared a priority last April.
Senator Patrick Leahy, a Democrat, pointed to Pompeo's denunciations of Venezuela's leftist President Nicolas Maduro as a "tyrant" and asked if he would use similar language for Kim.
The gathering of the Central Committee comes after Kim's Hanoi summit with US President Donald Trump broke up without agreement in February, and as South Korean President Moon Jae-in flies to Washington for talks with the US leader.
The exact date of the trip has not been set yet.
President Moon Jae-in was instrumental in brokering the negotiations between Pyongyang and Washington, seizing on the 2018 Winter Olympics in South Korea to catalyse a rapid diplomatic rapprochement after a year of missile tests, threats and tensions.
Early in March, several American think-tanks and South Korean officials reported that satellite imagery showed possible preparations for a launch from the Sohae rocket launch site at Tongchang-ri, North Korea.
This year's turnout fell just short of 100 percent as those 'abroad or working in oceans' were unable to take part, the North's official KCNA news agency reported.
US National Security Advisor John Bolton told ABC News that Trump's walk out from his meeting with Kim was "friendly" and it is not a reflection that the talks broke down.
The documentary broadcast on Wednesday shows a smiling Kim talking with Trump while walking together inside a Hanoi hotel last week.