While I agree that Batman should kill Joker there's a good reason why he shouldn't, take a look at their deeper, more philosophical, relationship, Batman wants to prove anyone can have an unbroken will and that anyone can get through hard times, Joker wants to prove that anyone can be broken and be forced to do something they wouldn't normally do, in fact Jason Todd's 'death' was meant to break Batman, it didn't, but it broke Jason, he survived and became Red Hood, now he holds an immense hatred for Joker for taking him away from Bruce, Red Hood was right when he said he's a better Batman than Bruce in that he kills people since they'll just break out of Arkham, but Red Hood is also the perfect example of what will happen to Batman if he kills

Joker wanted to break Batman so he tortured and 'killed' Robin (Jason) but Batman pulled through, not because he's an emotionless psychopath but because he's a strong willed person, Jason was so enraged and tortured that he broke, he knew how much Bruce was hurting emotionally so he tried killing as many villains as possible to make sure that something similar doesn't happen again, he had the same guilt that Bruce did, Batman has already seen what the outcome of Joker winning someone, they might not be evil or have evil intentions but Joker still wins

Even though I agree that Batman should kill Joker he absolutely can't, as I said previously Robin basically decided to kill villains because they deserve it (I agree on that) and even if Batman agrees also he still can't kill them, if Batman breaks than he won't stop, he'll keep breaking until he's no better than Joker, so while he won't begin evil he will slowly turn evil, just like Gordan(?) said in The Dark Knight "You either die a hero, or live long enough to see yourself become a villain"