The US Pacific Command confirmed the failed test had taken place on Saturday (16 April), with officials saying a missile had exploded within seconds of the launch.
"You read Clinton's book, and he said 'Oh, we made such a great peace deal.' And it was a joke", Trump said.
Mr Kim stressed that US-South Korean military exercises being staged were now the largest-ever "aggressive war drill" aimed at his country, formally the Democratic People's Republic of Korea.
On Monday, US Vice President Mike Pence-visiting the world's most heavily militarized border that divides the two Koreas-hinted that military options against the North have not been ruled out.
One reporter asked President Trump if he had any message for North Korea.
"I can see that the USA is also emphasizing and reaffirming their willingness for a political and diplomatic solution".
China may be getting fed up with continued nuclear bluster from long-time ally North Korea and tilting toward the United States.
North Korea's vice-foreign minister Han Song-Ryol told the BBC missiles would be tested on "a weekly, monthly and yearly basis".
Pence pointed out the recent actions of the U.S.in Syria and Afghanistan demonstrated the power of the us military and Trump's resolve.
Mr Mattis did not identify the type of missile but said it was not of intercontinental range, meaning it could not reach United States territory.
Pence's visit to the tense demilitarised zone dividing North and South Korea came at the start of a 10-day trip to Asia and underscored U.S. commitment to the region.
It's necessary to put pressure on North Korea so that it will accept dialogue in good faith.
At the outset of his meeting with Abe, Pence said that "While all options are on the table", Trump was determined to work with Japan, South Korea and other allies in the region to resolve the problem.
His comments came after a North Korean diplomat told a United Nations summit that thermonuclear war could break out "at any moment".
On Monday, the North's deputy United Nations ambassador, Kim In-ryong, told reporters that the USA has "created a unsafe situation in which thermonuclear war may break out at any moment on the peninsula". I hope things work out well. Trump said in an interview with Fox News. "And also seek their views on how best to address the North Korea threat", Klinger told CNBC's "Squawk Box" on Wednesday. He also said it was reassuring that Mr Pence had not cited trade reforms as a pre-condition for the US-South Korea military alliance. While Obama's strategic patience doctrine often receives bipartisan criticism, Lavrov questioned the accuracy of the label given previous US led sanctions against the DPRK.
The Trump administration has signaled a more forceful USA stance toward North Korea's recent missile tests and threats, including a warning from Trump that North Korean leader Kim Jong Un has "gotta behave".