Kim also expressed optimism that "the strong will, honest efforts and unique approach" by the two leaders will open up a "new future" between the USA and the North.
Mr Trump describes the letter as a "very nice note" and says, "Great progress being made!".
North Korea leader Kim Jong Un inspects Unit 1524 of the Korean People's Army (KPA) in this undated photo released by North Korea's Korean Central News Agency (KCNA) June 30, 2018.
Kim did not meet with Pompeo during his visit this month, viewed widely by foreign affairs analysts as an intentional slight since the two had met twice previously in Pyongyang.
The letter was heavy on flattering language but offered nothing towards denuclearization, which has been the US's stated objective in engaging with North Korea.
In another sign that relations have strained since Kim sent the letter, CNN reported that North Korean officials failed to show up for a meeting with their USA counterparts in the demilitarized zone Thursday.
The United States and North Korea have agreed to hold talks Sunday to discuss the repatriation of American troops' remains from the 1950-53 Korean War, a U.S. State Departmentspokeswoman said.
Pompeo insisted the talks were making progress but as soon as he left, the North's foreign ministry berated him over his "unilateral and gangster-like" demands and for offering no constructive steps on the U.S. side.
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But in stark contrast, Pyongyang's take was overwhelmingly negative, with the North warning that the future of the peace process was being jeopardized by overbearing U.S. demands for its unilateral nuclear disarmament. He added that "great progress being made!".
But while improving relations and decreasing hostility has certainly been a benefit of Trump's diplomatic push on North Korea, it's been a means to the end of denuclearization.
Dollar said that the institutions could start helping North Korea with technical assistance and training "even before they are members".
"He's realised he can humiliate and ignore the USA as long as he flatters the president", Mount said.
An estimated 30,000 United States soldiers died in the 1950-53 Korean war.
A source with knowledge of the discussions between Pompeo and North Korea said the feeling at the White House was the trip went "as badly as it could have gone".
Trump himself has remained upbeat about the outcome of the first summit between the leaders of the USA and North Korea.
However, critics have repeatedly said that Mr Trump has since failed to secure any firm commitments from North Korea toward the dismantling of its nuclear weapons.
During an official visit to Britain, Trump posted images of the Korean-language letter from Kim and its English translation. On the day he left the North issued a statement on state-run media accusing the United States of "gangster-like" demands for unilateral disarmament.
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