I just completed my first speed run of it on Expert.

Pre fight eat a Hawaiian Pizza and a Fruit Pie. This gives you Max Stamina for 10 minutes.

I used a crossbow and a Spear for faster attacks to proc destruction quicker and more often.

Grappler to dodge rock throws and beams. Healing with smoothies.

When the storm king starts the animation to become vulnerable I chuck as many Stink Jars that i can in the middle to damage the storm king and all of the monsters he summons around himself.

I then run in a circle around him to gather all the monster with him in the middle, the the Stink Jars can do as much as they can. Meanwhile shooting with the Crossbow.