Although Pompeo insisted progress continued to be made, North Korea accused the USA of making "gangster-like" demands.
"The significant first meeting with Your Excellency and the joint statement that we signed together in Singapore 24 days ago was indeed the start of a meaningful journey", the letter reads.
Kim said he hoped the process between the two sides would lead to "practical actions" in the future, though he offered no specifics, and that there would be a second meeting between the two leaders.
North Korean officials did not show up to a planned meeting with United States officials at the demilitarized zone Thursday, where they were expected to continue discussions on repatriating the remains of Americans killed during the Korean War, a senior U.S. official tells CNN.
According to President Trump, returning the remains of USA troops was agreed upon during the Singapore summit.
The secretary of state did not meet Kim as he had on his previous two trips to Pyongyang this year, and after his departure, the North's foreign ministry accused the US of making "gangster-like" demands that it unilaterally give up its nuclear weapons.
"Both the United States and the North are still in discussion over when to meet", the official added. "It is likely that military generals from the USA and North Korea will take part in the meeting".
The fact that North Korea seem to be dragging its feet on this issue fuels already growing doubts among President Trump's critics about whether the Singapore summit was a success or simply a publicity stunt for Mr Kim, the BBC's Cindy Sui in Seoul reports.
"We were ready", the official said.
Though Mr. Trump claimed on June 13 that North Korea was no longer a nuclear threat, just a day after his meeting with Mr. Kim, experts are not sure if North Korea is following through on the deal.
It was not immediately clear why North Korean officials had not attended the meeting or whether they had ever confirmed their intention to.
Although Trump may have been delighted by the letter, Twitter users weren't impressed.
The letter comes days after the North Korean Foreign Ministry released a statement accusing the US of "ganster-like" demands and trying to unilaterally pressure the country into abandoning its nuclear weapons.
The President released the letter in Korean with an English translation.
The Department of Defense estimates that North Korea is holding about 200 sets of remains from some 5,300 American military personnel believed missing in the country.