It all started when Kelly forgot to lock the door to his office, which he says he usually does.
The anchor then invited the family to introduce themselves: his wife is named Jung-a Kim and the children are Marion and James. They could be heard but not seen long after the door closed.
"So I hope they stop arguing".
"She was videotaping the interview so we could have a clip of it and the kids sort of escaped and I had foolishly not locked the door", said Kelly. "It was chaos for me!"
While Kelly initially declined the BBC's request to share the interview online, he eventually allowed the posting after the network had insisted it showed they were a normal family. In this case, the rest of the family had been watching him on the TV in their living room as he conducted the interview in his office.
"Everybody we know seems to think it's pretty hysterical, " he told the BBC. It went viral nearly instantly, reaching 84 million views on the BBC'sFacebook page alone, and getting media coverage worldwide. "We love our children very much, and we are happy that our family blooper - our family error on television - brought so much laughter to so many people". Kelly wore jeans with his dress shirt and tie, and Kim was filming the clip on her phone.
Mrs Kelly said she phoned her son after the incident, and he admitted to being a bit "embarrassed", though he did see the amusing side. "Yes, I was mortified, but I also want my kids to feel comfortable coming to me". But Kim Jung-A took it in stride - as did her husband - who added, "I hope people just enjoy it and don't argue over this thing".
Trump envoy meets Netanyahu in Israel
Envoy Jason Greenblatt arrived at the West Bank city of Ramallah where he was greeted by Abbas and a Palestinian delegation. Trump said he would be inviting Abbas to the White House in the near future.
Watch out Allegri, Wenger wants to stay at Arsenal
If Wenger was to leave, the club would accelerate their search so the new manager would be joined by the sporting director. Arsenal's French manager reaches the end of his current deal in the summer and has yet to commit to staying.