Once you've finished the main episode once, you'll unlock a new option on the menu allowing you to redo Chapter 3 but with the option to "play along" with Ardyn. Choose this option and you'll be able to fight Ardyn a second time, this time actually being able to defeat him.
It's a slow but easy fight as Ardyn has a lot of health, but keep hitting away and you'll get there. When the ring runs low on power, you want to select the "risk your life" option on the first ask, then on the second choose to sacrifice your life. This will start a timer but should be plenty of time in which to finish off Ardyn.
I am unsure if repeating the "fight back" option allows you to defeat Ardyn or if, like in the main episode, it ends the fight early before he has run out of health, but you'll want to pick the "play along" option to see the alternative storyline anyway.