Watched it yesterday, Batman didn't kill joker, ultimately bc what Gordon said at the end to batman and how Gordon himself didn't go insane. Batman/Joker dualistic relationship will always be present, like the opposite of attraction such as night and day, man and female, good vs evil etc. Batman sort of discussed this in the movie, and it's present in the mainstream blockbuster movie, Dark Knight with heath ledger, that batman and joker will always exist; they can't kill each other even if they wanted to. It's present in the comics too, batman is nothing without joker and vice versa.
The joke at the end is weird no doubt, but the ending sort of explains their relationship, they can't kill each other only the joke, the analogy of good and evil, sort of is the punchline to Batman and Joker and their creation. The joke is that both batman and Joker are insane in this cosmic universe of light vs darkness. How I took it when batman laughed at his joke, sort of sums up that batman's character is an evolving thematic consequence when you fight darkness for so long, you sort of become what you trying to destroy. The same thing is said in the Dark Knight, how you live long enough you become the villain.
Many people will take it that bc Batman is insane and he laughed at the "Joke," he ultimately kills joker and batman gets put in jail or dies, but I think why batman didn't kill joker is because of what Gordon says. I could be wrong, but at this point, who cares. Batman will always be the crusader of good against evil. Now and forever.