Hold L2, insta-kill only works on green enemies. Most palaces won't have green unless you are really over leveled.

How do you insta kill in palaces?

Insta-kill, rather than activating out of every stealth attack on low-level enemies, is now triggered by sprinting through the enemy instead. In addition to that, enemies that can be Insta-killed are highlighted in a new green outline when using the third eye ability in game.

What is the best palace in Persona 5?