The White House confirmed the summit had ended with “no agreement reached” as the leaders headed back to their respective hotels.
The US president talks broke down over North Korea’s demands on US-led sanctions.
“Basically, they wanted the sanctions lifted in their entirety, but we couldn’t do that,” he told reporters. “Sometimes you have to walk.”
Sarah Sanders, the White House press secretary, said negotiations would continue at a future date.
Several Democrats came out acknowledging Mr Trump’s decision to walk away without a deal was the right move in this situation. Democratic Congressman Adam Schiff said walking away with no deal was better than agreeing to a bad deal, before adding that it was “the result of a poorly planned strategy.”
Senate Minority Leader Chuck Schumer echoed similar statements, citing his concerns about the likelihood of a bad deal forming out of the summit.
“A deal that fell short of complete denuclearization would have only made North Korea stronger & the world less safe,” Mr Schumer said.
After the summit, Mr Trump also defended Mr Kim over the tragic death of American college student Otto Warmbier, who was jailed in North Korea in December 2015 for attempting to steal propaganda material during an organised tour.The president said he does not believe the autocratic leader was aware of Mr Warmbier’s condition in the North Korean hard labour prison camp.
"He tells me he didn't know about it, and I will take him at his word,” Mr Trump said.
After two years of imprisonment, North Korean authorities returned Mr Warbier to the US in a coma in July 2017. A few days later, the 22-year-old died in his hometown of Cincinnati, Ohio.
Ohio Senators Sherrod Brown and Rob Portman criticised the American president’s defense of Mr Kim.
“I’m very concerned that the President didn’t seem to be all that concerned about the murder of Otto Warmbier from Cincinnati,” Mr Brown told reporters on Thursday. “I don’t know how he says he likes the dictator of NK so much.”
Mr Portman insists that Mr Trump and the American people must remember Mr Warmbier and that “we should never let North Korea off the hook for what they did to him."
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Hello and welcome to The Independent's rolling coverage of the Donald Trump administration.
Today the Democrats begin their attempt to roll back President Donald Trump's declaration of a national emergency to siphon billions of dollars from the military to fund construction of a fence along the US-Mexico border.
Tuesday's vote in the Democratic-controlled House comes on legislation to revoke Trump's executive order from earlier this month and would send it to the Republican-held Senate, where it would take only a handful of defections to pass.
President Trump is likely to prevail in the end since he could use his first-ever veto to kill the measure if it passes Congress but the White House is seeking to minimise Republican support for the measure to avoid such an embarrassment.
The vote could be challenging for conservative lawmakers who view themselves as protectors of the US Constitution and the powers of the federal purse that are reserved for Congress. But the party's vote counters are confident the tally won't get near the two-thirds that would overturn a Donald Trump veto.
Democratic leaders said on Monday the vote is not about the merits of Mr Trump's wall but how the president is trampling on the Constitution by grabbing money he can't obtain through normal means.
"The beauty of the Constitution, the heart and soul of the Constitution, is the separation of power - co-equal branches of government to be a balance of power," said House Speaker Nancy Pelosi. "The Constitution spells out the responsibilities, giving the Congress of the United States, among other powers, the power of the purse. The president's power grab usurped that constitutional responsibility and fundamentally violates the balance of power envisioned by our founders."
"If Republicans vote their beliefs, we'll get a lot. If they vote their party, we won't get a lot," said Democratic House majority leader Steny Hoyer.
President Trump took to Twitter on Monday to urge Senate Republicans to stick with him:
Vice-president Mike Pence is expected to discuss the issue with Republican senators during their weekly private lunch. A Justice Department official is also expected to attend.
On Monday, North Carolina senator Thom Tillis became the second Republican senator, joining Alaska's Lisa Murkowski, to say he would vote to block the order. Congress must defend its power of the purse and warned that a future Democratic president might abuse the power to advance "radical policies," Senator Tillis said.
Senate voting on Donald Trump's emergency order could drag under a rarely used procedure, which is believed to be a first for the chamber.
The law allows for up to 15 days of committee review— in this case, at the Armed Services panel — with a full Senate vote three days later. Senators, though, said the process could be expedited.
At issue is President Trump's longstanding vow to build a wall along the 2,100-mile southwestern border, his signature campaign promise. He has long since dropped any pretense that money for the wall would come from Mexico, which he initially loudly claimed would be the source of funding.
Earlier this month, Congress approved a huge spending bill providing nearly $1.4bn (£1.1bn) to build 55 miles of border barriers in Texas's Rio Grande Valley, ending a dispute that had led to a record 35-day partial shutdown of the government.
The president had demanded $5.7bn (£4.3bn) for the project and the national emergency declaration grants him emergency powers to divert $3.6bn (£2.7bn) from military construction projects towards the wall.
Lawmakers in both parties are recoiling at the politically-toxic prospect of losing cherished projects at back-home military bases. The Defense Department has yet to identify, which projects may face the axe.
Yesterday, national security experts and former Republican lawmakers issued public declarations against Mr Trump's edict, saying that the situation along the southern border is not a genuine emergency and that Mr Trump is abusing his powers.
"We are aware of no emergency that remotely justifies such a step," wrote 58 former senior national security officials, including Republican Chuck Hagel, a former Nebraska senator and defense secretary. "Under no plausible assessment of the evidence is there a national emergency today that entitles the President to tap into funds appropriated for other purposes to build a wall at the southern border."
In addition, 28 Republican former House members and senators, many of them from the party's shrinking moderate wing, wrote an open letter declaring their opposition to Trump's emergency declaration.
"How much are you willing to undermine both the Constitution and the Congress in order to advance a policy outcome that by all other legitimate means is not achievable?" wrote the former Republican lawmakers, among them former Senator Richard Lugar, once the chairman of the Senate Foreign Relations Committee.
"It was a lawless act, a gross abuse of power, and an attempt to distract from the fact that he broke his core promise — to have Mexico pay for the wall," said top Senate Democrat Chuck Schumer of New York.
Mr Trump's edict is also being challenged in the federal courts, where 16 Democratic-led states such as California are among those that have sued to overturn the order. The House may join in.
The president is currently on his way to Vietnam for his second nuclear summit with North Korean leader Kim Jong-un, whom he hopes to persuade into abandoning the country's nuclear programme.
Here's The Independent's diplomatic editor Kim Sengupta on what to expect in Hanoi and whether it will amount to more than a mere photo opportunity.
BlacKkKlansman director Spike Lee has responded to Donald Trump's attack on him yesterday.
Accepting the statuette for Best Adapted Screenplay at the Oscars on Sunday night, Mr Lee told the audience: “The 2020 presidential election is around the corner... Let’s all mobilise, let’s all be on the right side of history, let’s choose love over hate, let’s do the right thing.”
To which the president responded:
The filmmaker has since told Entertainment Weekly, “Well, it’s okee-doke, you know. They change the narrative. They did the same thing with the African-American players who were kneeling, trying to make it into an anti-American thing, an anti-patriotic thing, and an anti-military thing. But no one’s going for that.”
Here's Jack Shepherd.
The Spike Lee controversy was not the president's only inflammatory comment yesterday, far from it.
Last night he criticised Democrats in the Senate for voting against anti-abortion legislation, calling it "one of the most shocking votes in history of Congress".
"If there is one thing we should all agree on, it’s protecting the lives of innocent babies," he wrote.
Kim Jong-un has arrived in Hanoi in high style - his limousine flanked by 12 jogging security guards - after travelling 2,500 miles across China by armoured train to attend his latest summit with Donald Trump.
Mr Kim's locomotive had earlier pulled into Dong Dang rail station on the border where it was met with a red carpet reception following the arduous 65-hour journey.
President Trump wrote this before departing:
Here's Chris Baynes.
Another of yesterday's dramas surrounding the president was the news he was being sued by former campaign staffer Alva Johnson, who alleges he kissed her without consent at a rally in Tampa, Florida, in August 2016.
The White House denied the accusation, branding it "absurd".
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