Alright, you inspired me to do a little research....
2012 - Kings: Did not go 7 games
2011 - Boston: Went 7 games in the 1st round and 3rd round
2010 - Chicago: Did not go 7 games
2009 - Penguins: Went 7 games in 2nd round vs Washington
2008 - Detroit: Did not go 7 games.
2007 - Ducks: Did not go 7 games
2006 - Carolina: Went 7 games in 3rd round vs Buffalo
2005 - Lockout
2004 - Tampa Bay: Went 7 games in 3rd round vs Philly
2003 - New Jersey: Went 7 games in 3rd round vs Ottawa
2002 - Detroit: Went 7 games in 3rd round vs Colorado
So just taking a relatively small sample size of the last ten NHL seasons, 60% of the teams that won the cup played at least one 7 game series before the Stanley cup final.
Hrm, for some reason I thought it was more rare, I stand corrected. I still feel like the Kings already are looking tired as hell, but who knows, anything can happen!