Why Didn't Super Saiyan Blue Goku Use Kaio-ken Against Broly?

The Manga

Well to put it simply, thinking outside of simply the story itself, Goku could never use Kaioken while in Super Saiyan Blue form in the manga itself. Goku mostly struggled to master the Super Saiyan Blue form, usually flipping between Blue and God forms in order to conserve strength and stamina. Dragon Ball Super: Broly was created by Akira Toriyama himself while the anime was produced by Toei Animation. With the anime creating this Kaioken technique, they did a worthy job of explaining just why Goku could use it in Blue but not in the traditional Super Saiyan form, but that wasn't enough for it to make the cut of the film.

Kaioken Is Dangerous

Going into the story reason, Kaioken is a dangerous move and I think that Goku's reliance on it throughout the series can sometimes make us forget that. King Kai tells the Saiyan protagonist as much when he first begins using it to a large extent during his battle against Vegeta, with the Kai losing his mind originally when Goku was willing to multiply it times four. We couldn't imagine that Kaioken times twenty would have even been enough to take down Broly and we have to think that it might not have been a road Goku wanted to go down.

Goku Thought It Wasn't Worth It

When Broly had outmatched both Goku and Vegeta in their one on one fights, then dismantling them when they teamed up, we're sure that Goku had two options in front of him. He could either put his effort into Kaioken and potentially draining himself of all his energy and stamina or take the road he did travel in the form of the fusion dance. Gogeta is a being that is far beyond any other character that we've seen before, excluding the likes of Whis and Zeno that is. Why would Goku rely on Kaioken when the fusion was an option?

There Wasn't Time

In both the story, and the run time of the movie, perhaps there simply wasn't enough time for Goku to whip out this trump card. The fight itself moved at a frantic pace and Goku moved from his one on one battle with Goku quickly into teaming up with Vegeta. Either Kaioken or Vegeta's Super Saiyan Blue Evolved technique might not have been options as Goku realized that the fusion dance was going to be their best option for bringing down the Legendary Super Saiyan!

Broly Was Evolving

Broly was a villain that hadn't been seen before in the main continuity of the Dragon Ball franchise, and the Legendary Super Saiyan was "learning" as he fought against both Goku and Vegeta. Each time one of the Saiyans transformed, Broly would have his power level grow exponentially in response to whatever the protagonists of the series threw at him. Perhaps Goku felt that using Kaioken while in his Super Saiyan Blue form simply wouldn't be able to overpower Broly fast enough for him to not overcome it.

The fusion of Gogeta was so far and beyond Broly's power, even following his ascension to Super Saiyan, that it left the Legendary Super Saiyan no chance of winning at the end of the day!