Mr Kelly, a political science expert, was interrupted by his two small children during a live-on-air Skype call on Friday.
Marion, four, barged into the room and danced in the background as her father discussed the South Korean government. She was followed by her nine-month-old brother, James, who rolled in on a baby stroller, and then their frantic mother, Jung-a Kim, who clattered into the room in hot pursuit of her escaped children.
Holding his baby son, Mr Kelly said in his first interview since the incident that he and his wife found the video just as funny as everyone else, but they were also mortified at the prospect of having "completely blown our relationship" with the BBC and worried that Mr Kelly might never be asked to speak on television again.
In interviews with the Wall Street Journal and the BBC, Mr Kelly said he watched the video with "a mixture of surprise and embarrassment and amusement... and I suppose love and affection, I mean it was terribly cute... it is really funny".
He also confirmed that despite speculation to the contrary — based on the fact he remained seated throughout the debacle — he was wearing trousers during the interview.
Ms Kim said the chaotic scenes were true to life, but usually the kids only ran amok in private.
"It happens all the time for us, but not in an interview like this... this was the first time," she said.
Mr Kelly agreed. "Yes, I've been doing TV for a long time and this has never happened before," he said. "Obviously."
"Yes" his wife retorted, "...because mostly you lock the door."
The couple, who live in South Korea, said that although they have had to turn their phones off due to being "relentlessly" hounded by the media and public since Friday night, the feedback they have received has been overwhelmingly positive.
"Everybody we know seems to think it's pretty hysterical," Mr Kelly said.
"We understand why people find it hysterical... Catching a regular family off guard and stuff."
But he admitted that they were both "pretty uncomfortable" with the widespread assumption that Ms Kim was the nanny, not the children's mother.
She said: "I hope people just enjoy it [the video] and don't argue over this thing... But I'm not nanny. That's not true. So I hope they stop [saying it]."