Because Prince Seon and his allies were completely incompetent at trying to solve the poison issue on their own, Hwa-goon's dad pops out of nowhere and just gives them the formula for the antidote on one condition. It should come as little surprise that even in his final dramatic victorious moment, Prince Seon manages to find a way to screw things up in this regard. No matter how simple the task, Prince Seon is just not up to the job.
Hilariously, even other characters have zero faith in Prince Seon, as the legions of Dae-mok's poison victims straight up refuse to try Prince Seon's antidote. Why? Because it will probably kill them since Prince Seon can't do anything right. What's especially ridiculous is that there's good precedent for this. Recall how the whole reason the other Seon turned against Prince Seon was because the last time Prince Seon procured an antidote it just turned out to be an even worse poison.