There is no reason to keep Kai from walking up to her and smacking her down in front of the entire palace. Which, in honesty, is his best option.
All he has to do is say 'You dare act in my name?' and it's over. She committed a crime. Unless she and uncle have armed soldiers on their side they would lose access to the palace.
Even if we suppose that Kai is too young to handle this nothing stops his supporters from doing so. Thier well-being depends on his image in front of the palace.
The next logical thing would be to purge the palace. So... we're in lala land now. Let's see how rules in this universe work.