Would Kim even need to do anything? Legally or ethically. She already knows the truth and the tape is nothing more than Jimmy telling his mentally unfit brother what he needs to hear so he will calm down and stop literally wallpapering his home with tinfoil. Chuck is too far gone, and too many people know it.
I can't see the tape being used on it's own, more the first piece of a case Chuck will start putting together next season to take Jimmy down. It also shows Chucks willingness to lower himself to Jimmy's level. But these are the Breaking Bad writers and like others have said they don't even know what they'll do. They love writing themselves into corners and having shitty payoffs. Fucking machine gun.