Dog eat Dog
If Batman were to kill his nemesis, its possible he'd suffer from an identity crisis. Batman was spawned from the man to become the Clown Prince of Gotham. Before I delve any further, I'm not claiming to be an expert nor criminal psychoanalyst. Just another humble opinion, a fan I am. For a playful parallel, think of the antics between the Looney Tune characters Ralph Wolf and Sam Sheepdog. Both are merciless in their duty to accomplish their goals. And in the same breath, merciless against each other. But in the beginning and in the end of the cartoon, it shows them both punching in and out of the timeclock as if it were to be done. They both exist for their purpose. And without the other, their purpose would less satisfying. Less meaningful. On a subconcious level, I think Batman needs Joker in his universe to keep his edge; he's the carrot that keeps him chasing.
Alternatively, maybe its not up to Batman. Joker always seems to end up in Arkham, where some lawyer or pyschiatrist wanted to spare him. Personally, I think it illustrates the institutional failure of Gotham politics to think caging the animal, even with an insanity plea, is better than putting the wild dog down. But this isn't what I concur with. Biggest part of my love for this criminal bromance is that it does remind me of the Looney Tune duels of old. And everytime I see a new Batman comic, game, movie being announced, I can always count on Mr. J to come popping from the shadows to flaunt his chaotic grin through the streets of Gotham.