Kokichi wasn't the mastermind? He wanted to trick Monokuma?

The thing that was more distracting you was the poisonous arrow in his back. And the way he was telling Kaito to end him.

"Don't tell me you wanna waste my death. Don't you wanna end this hell of a game too?"

his voice getting more and more weary from the poison. Still he manages to keep up that irritating smirk

He'll nag him till he dies if he has to. What he doesn't know- you're watching the whole thing