What you're neglecting is that Gotham's criminal underworld is a matter of public record in the DC Universe. Kira could simply Google search the real names of every single person serving time in either Blackgate or Arkham, and then eliminate the entirety of them within a few minutes. With one major exception...
Joker has no real name. So your scenario would begin with Joker suddenly finding himself as the last living criminal in all of Gotham. Joker would essentially become the evil 'L', as the only way for Kira to finish him would be to gain the eyes so he could see his real name, which Kira has been clear he's unwilling to do.
So how might Kira finish the deed? Well what if he released a video condemning the Gotham Police and Batman for letting Joker run free for so long, and in it he stipulated that unless Joker was caught and executed, he'd kill James Gordon... Giving Batman and the Gotham PD like 72 hours to get it done.
Ok, so now that Kira has gone public, L is lured in, but in a twist, Batman will seek out Kira while L, Nightwing, Batwoman, and the Gotham PD find Joker. This is because Batman knows that if he catches Joker, he will not be able to hold himself back from killing him if it means saving Gordon.
I could go on and on, but I figure that'd be a good start.