Walke to the main arena and start by focusing damage on the Second Prince, getting his health low is the key here dont try using the Kings attacks to damage the Prince that doesnt work well. Keep both bosses in your view and use the space smartly, dodging the Princes quick dash attacks right at the last second gives you good windows to counter attack.

When you get those openings use your transformation powers for big damage then switch back to save spirit energy maybe interesting that killing the King first gets you the Second Rat Prince spirit which helps stagger enemies when youre low on health but honestly its easier to take down the Prince first since the King runs away after. The Prince gets super aggressive after the Kings dead so save your strongest transformation for that part and use the arena for cover while you focus on dodging and countering his combo attacks.

The Ebongold armor set (or Serpentscale Battlerobe) helps with defense, and keeping your dodge at medium weight is really important for staying mobile while having enough protection to survive mistakes. Watch the Princes movements carefully and dont get greedy with attacks since both bosses can combo you down pretty quick if youre not paying attention tip that using the pillars around the arena can give you some breathing room when you need to heal or recover spirit energy.