For starters, the French economy would be completely wrecked by the ransoms they would have to pay. At that point, the English could really make some extremely strong demands and could possibly even demand that French crown renounces their overlordship over Gascony, Calais, and any and all other territories they so wish. Keep in mind that King David of Scotland is also in captivity, so they could also demand the dissolution of that pesky Franco-Scottish alliance, French renounciation of any and all claims to the Low Countries, and the return of ancestral Norman and Angevin lands previously lost. I don't see how France possibly recovers from this. This would be an utter catastrophe, even more than if King John and all his sons were killed in battle. That at least would save the economy.
Oh, and to say the least, the House of Plantagenet taking the French Crown. This is screw-turning time if they really want.