Well, they both hear about it at the same time, and they know the other will be gunning for them. So, however long it takes for one of them to decide to strike first. I could say at least a day or something, but I'm not sure if they would both wait that long, or move that early, pending why they'd be fighting. They both have the same amount of information, and one of them is going to move at some point, and that doesn't necessarily mean a day or 2 days or a few hours at all.
Luthor should win just about every time. Joker's no dummy, but Luthor's possibly the smartest man on the planet and he has a battlesuit strong enough to put a bit of a hurting on Supergirl.
It has happened before when Joker wasn't as prepared as Luthor and Joker still won. It's not a matter of intelligence, Joker is just so random. I'd say Joker wins 5.5/10. He doesn't win a lot, but I think he would win.
Joker wins, he is just too insane. Luther is smart, but doesnt have the nerve to stick out a tough fight. Luther is more of a munipulator then a fighter. Luther would try to lure joker into a trap and if it failed he would high tail it at the first sign of real danger.
Yes I realize I keep spelling Luthor Luther. I have always had a problem with that, and the more practice I get with spelling Luthor on this forum, the easier it will get.