For the first battle, just do the normal. Stack up several status ailments on Kenny. Keep attacking him until he is defeated. After the first round, Kenny will drink the green goo, transforming him to a nazi zombie. Make sure to block his rainbow attack. It deals heavy damage even when blocked. When he's about to be defeated, (Princess) Kenny will attempt to drink the goo again. Your buddy will make a move to stop it so press the displayed key until you successfully stop his attempt.
Your buddy will leave and another one will swap in after this. Keep attacking like before and Kenny will keep reviving. Wait until he uses a special attack. Mash the displayed key to prevent the attack which will make your current buddy sit the rest of the battle out. Another one will come in and you have to kill Kenny once more.
Once a scene is triggered where Cartman grabs (Princess) Kenny, use a magic fart attack to take him out once and for all.You'll be back in your room the next day and congratulations for finishing the game!