Big argument arose last night as we were playing at the kitchen table. Clarification?

Player 1 holds 6,9 (suits unimportant)
Player 2 holds A, 4

Board shows K,K, Q,Q,6

Here's the conflict:
Does Player 1 or Player 2 win?

Player 1 maintains that the 6 he holds pairs the 6 on the board
Player 2 argues that the Ace is the kicker, since the two best pairs are on the board.