The Kings (est. 1967) are the 4th NHL team to win their first-ever Stanley Cup, so far, in the 21st century. Along with the Kings, the Anaheim Ducks (2007), Carolina Hurricanes (2006) and the Tampa Bay Lighting (2004) also won their first Cup since 2000.
So now the unwanted honor of being the oldest franchise WITHOUT a Stanley Cup title belongs to the St. Louis Blues (est. 1967) who came into the league the same year as the Kings and are yet to win a Stanley Cup. It'll be 45 seasons and counting for the Blues at the start of next season. The Buffalo Sabres (est. 1970) and the Vancouver Canucks (1970 NHL expansion team) are next with 42 seasons and counting.
For the record, the Toronto Maple Leafs, themselves, haven't won a Stanley Cup in 44 years. Their last Stanley Cup came in 1967-68, the year the Kings and Blues came into the league. But nonetheless, they do have a few Cups under their belt, so that leaves the Blues as the oldest Cup-less team.
As for other sports--
In the NBA, the Dallas Mavericks (2011) and the Miami Heat (2006) are the only teams to win their first-ever championship since 2000. Coincidentally, they did it against each other. If you're thinking Oklahoma City Thunder, no, the Thunder franchise actually won a championship in 1979 as the Seattle Super Sonics, so there won't be a first-time champion in the upcoming NBA Finals.
In MLB baseball, only the Los Angeles Angels of Anaheim (2002) and the Arizona Diamondbacks (2001) won their first World Series titles so far this century.
In the NFL, the New Orleans Saints (2010), Tampa Bay Buccaneers (2003), New England Patriots (2002), Baltimore Ravens (2001) and the St. Louis Rams (2000) were first time Super Bowl champions this century.
I guess the moral of this post is-- have patience sports fans, if your favorite team has NEVER won a championship, sooner or later, it'll happen. Well, if you live long enough.