Trump's almost admiring remarks came amid soaring tensions with North Korea over its missile and nuclear programs, with an alarmed Washington looking to China for help in reining in Kim.
Trump said he had "no idea" whether Kim was sane or not, but said the North Korean leader had faced a formidable challenge in taking over the country at a reported age of 27 after his father's death in 2011.
"He's dealing with obviously very tough people, in particular the generals and others. And at a very young age, he was able to assume power," Trump said in the interview with CBS's "Face the Nation."
"A lot of people, I'm sure, tried to take that power away, whether it was his uncle or anybody else. And he was able to do it.
"So obviously, he's a pretty smart cookie," he said.
"But we have a situation that we just cannot let -- we cannot let what's been going on for a long period of years continue," Trump added.
South Korea said the latest test was a failure. Trump refused comment on whether the United States had anything to do with the test failure.
"It is a chess game. I just don't want people to know what my thinking is. So eventually, he will have a better delivery system. And if that happens, we can't allow it to happen."
Hours before the North Korean test, US Secretary of State Rex Tillerson warned of "catastrophic consequences" if the international community does not act more forcefully to sanction Pyongyang.
The United States has deployed a naval strike group to the area led by the carrier USS Carl Vinson, which on Saturday began drilling with the South Korean navy.