"As soon as she opened the door I saw her image on my screen," said Kelly, speaking to the Wall Street Journal.
The 4-year-old bounced in, jubilant having celebrated her birthday earlier that day.
"She was in a hippity-hoppity mood that day because of the school party," he explained.
Kelly tried to get his daughter out of view, hoping that the BBC might cut away and give him time to usher the 4-year-old out of the room.
But they kept on rolling and, as Marion sat down, James was making his entrance - following his sister, and gliding in on his baby walker.
"Then I knew it was over," said Kelly.
Kim, still watching the television in the next room, had a delay on her set and did not immediately realise what had happened. Then, horrified, she dashed in to try and usher the children out.
"It's a comedy of errors," said Kelly.
After the interview had ended: "We said to each other, 'Wow, what just happened?'"
He immediately wrote to the BBC to apologise, but within 15 minutes the broadcaster asked if it could put a clip of the interview on the internet.
The couple initially declined, they told the newspaper - feeling uncomfortable that people might laugh at their children. But they were eventually persuaded that the video would show they were just a regular family.
The family soon realised what an impact it was having, with their phones ringing off the hook and offers from major US television networks flooding in. Some journalists tracked down Kelly's parents in the east side of Cleveland to ask them about it.
"We stonewalled because we didn't know what to do," he said.
The children, he insisted, were not in trouble.
"I mean, it was terribly cute," he said. "I saw the video like everybody else.
"My wife did a great job cleaning up a really unanticipated situation as best she possibly could. It was funny.
"If you watch the tape I was sort of struggling to keep my own laughs down.
"They're little kids and that's how things are."
Kelly and his family plan were holding a press conference today at his university to answer questions from the Korean media, which have a strong interest in the video.
"Yes I was mortified, but I also want my kids to feel comfortable coming to me," he told the paper.
"I made this minor mistake that turned my family into YouTube stars. It's pretty ridiculous."