I have K8 the other player has K9 K on the board and A comes on the river, who wins the hand
The winner is whoever can find the best five cards from among their private cards and the shared cards. We often describe a hand as fewer than five cards - if I have KKA92 I might say I have a "pair of kings" or a "pair of kings with an ace kicker", but the truth is that I have a pair of kings with an ace kicker and then a 9 kicker and then a 2 kicker; it's just that those later cards usually don't matter. In your case though, it sounds like they do matter - if the best cards you can find are say KKA84 and your opponent can find KKA92, you lose.
[Note that the best five cards don't have to use your own cards at all, so if e.g. the center has AAAAK then your Q8 would tie with an opponent's 72 because both of your best five-card hands are AAAAK.]
I'm assuming no "special" features such as the unnamed cards giving anyone a flush, or full house.
You will have a pair of kings. So does the other player. You both get the ace.
So far, your "hands" are both KK A.
You have an 8, and your opponent has a 9. At this point, you are trailing with KKA8 versus KKA9.
There will be a fifth card, X, to fill your hand. You have KKAX8, and your opponent has KKAX9.
Unless "X" happens to be a second 8, giving you a second pair, you will not win, because your 8 cannot beat his 9. In fact, you will lose unless,
The other three cards than the A on the board are something like QJT, or QQT, basically, three cards higher than your opponent's 9. Then you both "play the board," and his 9 versus your 8 becomes irrelevant, and you "tie."
If the 9 versus 8 matchup comes into play, you lose.