Best part is, there's a chance that having a pastel-yellow/pink colored magical talking pegasus pony simply ask him to play nice would work. Because that'd crazy enough that the Joker would have a hard time topping it. Of course, Fluttershy might be unavailable.
Seriously, though, I agree that a permanent Joker solution will involve Batman in one way or another. Try to kill Joker and Batman will stop you. But the only one who can catch Joker when he has escaped is Batman. Either way, it all begins and ends with Batman. So the first step is to get in touch with the Bat, unofficially, of course. It doesn't take a genius to figure out that Batman has friends in the police force. He's tolerated, and most likely has several contacts on the inside. (We know it's Commissioner Gordon, but that's not common knowledge in Gotham. Even if it is, I'd not tip my hand that I know.) Set up a meeting with the Commissioner and ask him about Batman. Suggest you want to meet with Bats soon to talk Joker, and let time pass for a day or so. One evening, you'll turn around and there's Batman.
Then go with the idea that Gotham might need a supermax style jail, or at least somewhere to house convicted supervillains like Joker, Mr. Freeze, Clayface and so on, but admit that I don't have much experience in what it'd take to keep those locked up. Ask Batman if he could give recommendations as to how you keep someone like Joker locked up, including not having him take over the place.
I don't expect Batman to be that helpful, though, and since I don't know who he is, I obviously can't expect Waynecorp to fund anything. Still, plant the idea that the Mayor would be willing to help with the supervillains by building a supermax jail and that, unofficially, I like what Batman is doing. I'm sure at this point, the next time I blink, he'll be gone, but that's okay. This has no easy solution, but letting Batman know that I'm willing to try anyway will hopefully put me on good terms with the vigilante. And get me a couple dozen Bat-bugs planted in my office and home. Batman's paranoia is legendary, after all. I'll pretend I don't know.
What happens next is the dangerous part, announcing the construction of a supermax jail to house the city's most dangerous criminals. Gotta force the Bat's hand a little here. This will obviously make me a target by a lot of the supervillains, but it should also gain me a guardian Bat, with a second visit following shortly after. Then I will suggest that Batman helps design the jail, or at least come with recommendations of what is needed to keep especially the Joker in jail, full well knowing that no matter what, Joker will probably escape once in a while, but at least keep the escapes a lot rarer.
Then all that's left is to get the jail built, position myself as "tough on supercrime" for political spin and wait for Batman to haul in the Joker again for his new personal room in the new superjail.
In theory, this could work. It will take a few years, but it's probable.