Due to how Ragnarok works with Death Lotus, at 6 (provided you're above 40% health or so or something), Olaf is basically impervious to Death Lotus. Now I wouldn't suggest going mid with Olaf, but in general, you can walk through Kat's ult if she's not fed and splatter her pirouetting ass against the wall.
Tristana can go toe-to-toe with Kat early game; if she Shunpos onto you as Trist, Explosive Shot and shoot her back. This is also a general mode of operation when playing against Kat: if she tries growing some balls and Shunpoing up to you, harass her back! She'll be a lot less keen to do anything when you've showed that you're not to be ****ed with. At 6, Tristana has Buster Shot to knock her back during Death Lotus and Rocket Jump to hopefully get away in time.
Vlad can sort of trade blows with Kat early game. If you're both 6, you have Sanguine Troll to get out of Death Lotus, provided you didn't blow it earlier. Afterwards, do the ol' Vlad Shuffle: Hemoplague, Ignite, Transfusion, whatever the hell it is that Vlads do.
Udyr's also great, although as a general counter rather than 1v1'ing at mid. Bear Stance allows you to slap her out of her ult, Turtle can mitigate a fair amount of damage, and if you're covering a lane and she's getting up in your face, just smack her back. You're Udyr, you can do that.
The other counters have been mentioned quite a bit in this thread already.