Gain increasing amounts of AD or AP, Adaptive over the course of the game.
Fleet Footwork on Kai'Sa helps you to sustain off some of the incoming harass damage during the laning phase and the extra burst of Movement Speed can be critical in positioning properly when diving for a kill/duel.
Triumph extra bit of burst healing mid-fight empowers your ability to dive an enemy to duel and make sure you get out alive or are healthy enough to keep fighting.
Kai'Sa's damage heavily relies on her ability to attack quickly and build passive stacks to shred the enemy and the bonus Attack Speed from Alacrity will go a long way in empowering that.
WhyCoup De Grace
Coup De Grace's execution damage help you to shred the enemy backline target's quickly and seamlessly so that you're able to get out and move on for more damage.
Kai'Sa has a naturally low attack range and is sensitive to being able to kite and maneuver around her opponents to deal damage from a safe range.
Celerity's bonus Movement Speed will help you to stay safe and grant you a bit of extra AD while your [E] is active.
Gathering Storm is your late game damage insurance policy, allowing you to scale and take over fights/duel any in a 30min+ game.