The song became an adopted anthem of Welsh football fans during Speed's playing career with Wales after being used in a BBC Wales promo for the 1994 World Cup qualifying campaign.
Speed, who was found dead at his Cheshire home aged 42 on Sunday, was an integral part of the Wales squad that came so close to qualifying for the World Cup finals in the United States.
Jones broke off from working on new material with the band in Brussels to record his acoustic version of the song.
The video will feature as part of a special tribute to Speed in BBC Cymru Wales' Sport Wales programme on BBC Two Wales on Friday (2200 GMT).
Kelly Jones (a Leeds supporter): "I was with my two daughters at the Millennium Stadium after the Wales v England international [in March].
"We were waiting at the lift and Gary ended his phone conversation and shook my hand. We had a chat about the game and shared the lift to the ground floor.
"He was a very nice man. Good-looking fella. I was consumed the whole day when I heard the news. I couldn't get my head around it.
"Then the BBC approached me to record the song. It's a 10-minute job but I hope it's fitting for the tribute on the show."