North Korea watchers downplayed the significance of the letter.
In the translated letter, dated July 6, Mr. Kim praised his relationship with Mr. Trump and said the Singapore summit was "indeed the start of a meaningful journey".
The letter was dated just one day before North Korea accused the US of "gangster-like" demands following a visit and talks by Secretary of State Mike Pompeo.
Pompeo traveled to Pyongyang for two days last week in a bid to flesh out denuclearization commitments made during last month's historic summit between Trump and Kim.
There was no call or explanation from North Korean officials on why they skipped the meeting.
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The lawmaker's comments come one-day after North Korea criticized the USA for their push for denuclearization. When a reporter asked about North Korea's Saturday comments, Pompeo didn't seem too bothered.
"We have to see the USA defense ministry's response, but the UNC reportedly gave a positive signal to North Korea, so there is a good chance of the talks taking place on Sunday", the source said. "We will be ready", she said in a statement.
It wasn't immediately clear what prompted the postponement.
Last week, North Korea accused the United States of using a "gangster-like" tactics to push it towards nuclear disarmament after a fresh round of high-level talks. The talks Sunday are expected to involve officials from the Pentagon and the U.N. Command, which commanded US -led allied forces during the war and is involved in maintaining the armistice that ended the fighting in 1953.
It had been expected on Thursday at the inter-Korean truce village of Panmunjom.
But Pompeo gave a rosier account saying the two sides had substantive discussions on next steps toward denuclearisation.
The United States requested the UN sanctions committee to declare that North Korea had violated the UN-approved quota and "order an immediate halt to all transfers of refined petroleum products" to North Korea.
"Wishing that the invariable trust and confidence in Your Excellency Mr. President will be further strengthened in the future process of taking practical actions", wrote Kim.
But there's growing skepticism in Washington over the Trump administration's engagement with North Korea. Recent evidence actually suggests that the rogue nation continues to build up their nuclear infrastructure.