Why does this player not have a straight?
The problem is that a straight cannot be made with an Ace in the middle of the straight. Ace can be considered either the lowest card (forming a straight with A, 2, 3, 4, 5); or the highest card (forming a straight with 10, J, Q, K, A). It cannot be both low and high at the same time; so J, Q, K, A, 2 is not a straight.

In the case of this hand, player 1's best hand is A,K,Q,J,6; which doesn't form any combination; it is simply an Ace-high hand.

https://wiki.c2.com/?PokerGame


  The ace is the highest-ranking card in the deck, followed by the king, the queen, the jack, and the numbered cards in order. The 2 is the lowest-ranking card. In some hands, however (such as the straight and straight flush), an ace can be used as a "1" card - Ace, 2, 3, 4, 5 is a valid straight, as is 10-Jack-Queen-King-Ace. The ace cannot be in the middle; one cannot form a straight with Queen-King-Ace-2-3.


Another source, since there is no one difinitive rulebook for Poker to cite:


  The best possible straight in poker is the Ace-high straight (TJQKA) often referred to as “broadway”. The Ace can also be used to formulate the lowest possible straight A2345, often referred to as “the wheel”. Note however that it is not possible for the Ace to appear in the middle of a made straight. I.e. QKA23 is actually just Ace-high, not a straight. 


Note that you never get to choose "how many" hole cards you play. You always play a total of 5 cards; whichever 5 of the 7 options give you the best hand. IF player 1 had been holding a 10 instead of a 2, then player 1 would automatically play 10, J, Q, K, A; and would win the hand because of the straight, playing 1 hole card as you suggest.