This very convincing Reddit theory argues that Batman's sudden lax attitude towards killing is due to Batman having a mental break, brought about by his second Robin becoming the Joker. That's right, it's possible that Jared Leto's character was once Batman's apprentice, but was tortured to madness by a bad guy (the original Joker?). Let's look at some facts:
In the film, Robin's costume was marked with comments from the Joker, suggesting that the clown prince of crime killed him. But maybe his death was only metaphorical because where Robin died, the Joker was born? You know, like how Obi-Wan insisted Vader "murdered" Anakin Skywalker.
And then spray painted all over his Jedi robes.
After all, Batman does comment on how few people stayed good after 20 years in Gotham. That level of cynicism can be expected after watching his apprentice morph into everything he hates. Also, it does sort of look like Leto's Joker has tattoos covering up the spots where Robin's armor was shot (right shoulder, below the ribcage).