It had been expected at the inter-Korean truce village of Panmunjom.
"I deeply appreciate the energetic and extraordinary efforts made by Your Excellency Mr. President for the improvement of relations", Kim said, according to the letter, dated July 6. A total of 7,697 personnel remain unaccounted for from the Korean War. Soon afterwards the USA sent about 100 coffins to Panmunjom to put the remains into, but they have been sitting there ever since.
The remains of some United States soldiers were last returned in 2007, when then-New Mexico Governor Bill Richardson visited Pyongyang.
According to estimates from the Pentagon, 7,700 US troops are still unaccounted for from the Korean War.
The level of the meeting Sunday was not immediately known.
North Korea was expected to hand over those remains - believed to belong to some 200 to 250 United States servicemen - in the weeks following the meeting.
He praised officials of the county, which is close to Mount Paektu, the spiritual home of the Korean people and claimed by Pyongyang's propaganda to be the actual birthplace of his father and predecessor Kim Jong Il.
Pyongyang has already sent back the remains of 200 US troops, and the director of a USA defense agency said he is hopeful improved relations with North Korea will result in more recoveries. He said Monday it was "ludicrous" to expect results in a few hours of talks.
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"A very nice note from Chairman Kim of North Korea", Trump said. Pompeo's trip made no visible progress towards either of those goals, but produced a conspicuous snub from Kim, who preferred his potatoes.
The president says he believes the North Korean leader wants to denuclearize, and that he "trusts" him.
State Department officials had said that the meeting would likely take place on July 12, though they added that the date could shift. He also told supporters in Nevada that North Korea had already handed over 200 sets of remains.
"They didn't give it, I have it for him, they didn't give it, but it will be given at a certain time", Trump said when asked about the CD during a gaggle outside the White House Tuesday morning before departing for the North Atlantic Treaty Organisation summit.
North Korea watchers downplayed the significance of the letter.
North Korea has returned the remains of some U.S. soldiers to the United States over the years since 1953, when an armistice halted the Korean War. The North Korean foreign ministry described the talks as "cancerous" and accused Pompeo of using a "gangster-like" approach while negotiating terms to denuclearise.
At the Singapore summit on 12 June, Mr Trump and Mr Kim signed a document that included a pledge from North Korea to rid the Korean peninsula of nuclear weapons.
Satellite imagery appears to show a that nuclear weapons manufacturing facility in the secretive country has been strengthened and improved in recent weeks.
Trump touted progress between the two countries, but the note comes amid concerns over the feasibility of North Korea giving up its nuclear arsenal.
"Actual progress requires supporting the negotiators working to hammer out an agreement, including our allies". Defense Secretary Jim Mattis has said the UNC would initially take custody of the remains and preside at the dignified transfer to the USA military.