Trump shared the letter in a tweet Thursday, writing, "A very nice note from Chairman Kim of North Korea".
North Korea accused the United States on Saturday of making "gangster-like" demands in the talks in North Korea, contradicting Pompeo who said the old enemies had made progress.
In his letter, which was written 24 days after the June 12 summit in Singapore, Kim addressed Mr Trump as "Your Excellency Mr President".
Kim refers to Trump as "your excellency" five times, praises the "improvement of relations" between the USA and North Korea, and expresses hope for a new future.
"I deeply appreciate the energetic and extraordinary efforts made by Your Excellency Mr. President for the improvement of relations between the two countries", the letter read, in part.
Kim committed to "complete denuclearisation" when he and Trump met in Singapore last month.
Pompeo 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 USA of making "gangster-like" demands that it unilaterally give up its nuclear weapons.
North Korean officials didn't show up for a planned meeting with US counterparts to discuss returning the remains of American war dead, South Korean media reported, in the latest sign of tensions between the two sides.
Between 1996 and 2005, some 33 recovery operations were conducted in North Korea which saw 200 sets of remains returned. A confidential U.S. report presented to the United Nations security council, however, estimated the delivery of at least 759,793 barrels of oil products to North Korea between 1 January 1 and 30 May.
It said this had been done through ship-to-ship transfers at sea using 89 North Korean tankers.
The translated letter began, "Your Excellency Mr. President".
The committee was expected to take five days to consider the request, which China and Russian Federation are expected to block.
In another development on Thursday, North Korean officials failed to turn up for talks with a U.S. team on the repatriation of the remains of American soldiers killed in the 1950-53 Korean war. He later described his meetings with North Korean officials as "productive", while acknowledgnig that a lot of work remained to be done.
The US later said that North Korea had offered to meet on 15 July.
"The road ahead will be hard and challenging and we know that critics will try to minimize the work that we've achieved", he said.