Career
Jönsson was drafted by the Toronto Maple Leafs in the 1st round as the 12th overall selection in the 1993 National Hockey League Entry Draft. After playing with the Toronto club during the 1994-1995 lockout shortened season, he was traded midway through the 1995-1996 season to the New York Islanders in a player exchange which brought highly popular former captain Wendel Clark back to Toronto. Jönsson then spent the remainder of his National Hockey League career with the Islanders.
After playing with the Swedish second division team Rögle BK, where he started his career, during the 2004-2005 National Hockey League lockout, he announced in the summer of 2005 that he was staying with the club for the 2005-2006 season.
Jönsson played with the Islanders from 1996 through 2004. Fans and journalists considered Jönsson one of the most underrated defensemen in the league during his time in the National Hockey League. He played good positional defense and had sound offensive skills as well, but his quiet demeanor and low-key playing style kept him below the radar of some observers.
During the 1999–2000 National Hockey League season, Jönsson was named captain of the Islanders. After a season and a half, he stepped down and ultimately was succeeded by Michael Peca.
This was quite surprising considering he had spent a season in the Swedish second division, while most of the other players played in the National Hockey League. During IHWC 2006 World Championship, with brother Jörgen initially sidlined by injury, he was named captain of the Swedish national team
Prior to the 2010-2011 season, Jönsson signed as an assistant coach of Rögle BK. However, after the end of the season, it was announced that he would leave Rögle as an assistant coach. Kenny is currently assistant coach for Helsingborgs Hockey Club, educating the defense players.