It's going to be interesting to see what happens. As the others said Nick De Marco is the top dog for all these things but there is an added issue I can see Leicester having.
When Newcastle employed De Marco it was as a reaction to the Premier Leagues "new" rules and trying to see if they were legally enforceable (how our owner was defined for the fit and proper person test and then all sponsorship rules that were brought in after we were purchased). Although Newcastle "won" both cases the onus was on the league to show not only that their tests were legally correct but also that they were reasonable (boring bit: under administrative law you not only have to have the power to make a law and have followed the correct process (substative and procedural ultra vires) but you also have to show it was reasonable to do so - just because you can make a law doesn't mean you should).
Essentially, it was up to the league to show they should have done what they did. They failed to do so convincingly. The same is true of the Man City charges (To my understanding) at the moment. Before they look at whether Man City breached the rules the first thing is to establish if the rules should have existed and that's up to the league to argue.
With Leicester the laws being questioned are already established and have been enforced a number of times. The league doesn't have to show any of the above, it is up to Leicester to show they didn't breach these rules, or that the rules shouldn't apply to them, and I assume the league(s) believe they have compelling evidence that they did and they do. I'd have thought that was much harder to wriggle out of, last time they relied on showing the rules didn't actually apply to themselves on a technicality (there was no suggestion the rules shouldn't exist or that they hadn't breached them). Honestly, I'd be very surprised if they wriggled out of it again but if anyone can do it then it's Nick De Marco.