Texas holdem who wins
In Texas Hold'Em you use the strongest five-card hand that can be made using your two in-hand cards and the 5 board cards.  The two cards that you do not use are ignored completely and have no effect on the hand rankings.  (This is similar to 7-card stud and other 7-card variants)

In the example you present, it sounds like each player has an identically-ranked straight as their best five cards.  The pair of sixes would be ignored, as would the high-card king.  This would be a tie and the two players would split the pot.