The Black Prince's chevauchée earlier that year had incredibly damaged the south of France putting it, essentially, out of action. Now after Poitiers the English had the upper hand in the 100 Years War, a crushing victory which led to them capturing the French King and putting France into chaos.
It seems odd that with their enemy on their knees the English would not do their utmost to end the war there and then with a swift blow. Instead the Black Prince retreated to Bordeaux, and his father King Edward III didn't attempt to take Paris until 1359, which was unsuccessful.
But why? Did the English not have the will to do so, or did they still lack the strength of numbers?
They didn’t do that because it’s not really how medieval warfare really worked. There are a number of important factors to consider that influenced both medieval warfare generally and the specific circumstances of the Hundred Years War. I’m going to break down my answer into a few big points that I think are important to understanding why things developed the way they did.
France is big and, bear with me here, full of French people. Throughout the entire Hundred Years War England was fighting from a position of population disadvantage. Now, England actually had quite a large population for its size, but even if it had managed to have the same population as France it would still have fought from a disadvantage – after all, it’s not like the entire population of England could just pick up and move to France for an extended occupation, someone had to stay home.
This meant that wholesale occupation of France was never a viable option for the English king, so that meant that he had to win some hearts and minds. He needed the French people to accept him as their ruler, ideally as the King of France - replacing the existing King of France. As any student of modern warfare will tell you, this is easier said than done. The chevauchees, while devastating militarily, were not exactly popular with the local populace, and in many ways ran entirely contrary to a strategy of establishing Kind Edward as the rightful ruler of France. They did successfully undermine and weaken the rule of the existing French King, but they didn’t exactly make Edward the people’s (or noble’s, as they were the individuals with an actual say in a lot of this) choice for the next king. As a point of comparison, while Henry V initially started his military career doing these same kinds of raids, when it came to his conquest of Normandy he actually made significant (but not always successful) attempts to reign in the violence of his armies and try to preserve the people of Normandy so that he would seem like a powerful protector rather than a ravaging foreigner.
It is worth noting that the English king wasn’t an entire stranger in the areas of France where the Hundred Years War took place. He was, at least before the start of the war, the Duke of Gascony and throughout the Hundred Years War Gascony would express pretty consistent loyalty to the English crown – the generally accepted theory is that the Gascons were happier with a distant monarch who didn’t interfere with their affairs very often than they were with a local one who might meddle. It also bears noting that Gascony was still part of the Kingdom of France, which made the King of England technically a vassal of the French King which was a relationship that never went well.
However, Gascony was not the only area of previous English control. Obviously, the English monarchs themselves had roots in Normandy – William the Conqueror famously being Duke of Normandy – and they had, from Henry II’s marriage to Eleanor of Aquitaine, previously ruled much of north-western and central France. Now, basically all of these lands were lost under King John over a century before the outbreak of the Hundred Years War, but it wasn’t like this history was forgotten and the idea of the King of England ruling these territories wasn’t entirely farfetched.
In essence, though, to turn military victories into the battlefield into significant gains in physical territory the King of England and his generals needed to convince the local nobility (and to some extent the local populace, although they mostly didn’t care who was in charge so long as taxes were tolerable and nobody tried to kill them) that he was better at being in charge than the current guy was. This is a difficult thing to do and is often better managed slowly. If Edward III had managed to seize Paris, it’s not like the nobility all would have jumped to his side. Paris was an important city, but it wasn’t the end all and be all of French power. Nobles liked to be on the side that was going to win, and, if possible, that meant trying to play both sides against each other so that everyone thought you were on their side.
It is also worth remembering that south-eastern France has still largely unaffected by the ravages of war and so represented a source of potential resistance and soldiers for the French crown, meaning that Edward had a lot to gain from consolidating power in the regions he controlled than he did in sticking his neck out by pushing further into new territory. As a point of comparison, Henry V did succeed in conquering Paris, had himself crowned heir to the throne of France, and after his sudden death the French were able to push the English out of France even after all those considerable gains.
There’s a good few reasons why Edward III wouldn’t just have had Jean II executed. Some of them are moral – it was not the done thing to execute kings, at least not publicly – but many of them are deeply practical.
The first, and probably most straightforward, reason to not kill Jean II is that if Edward did then Jean’s son, the future Charles V, would just become king instead. Even if Charles couldn’t immediately be crowned the king, you’d just end up with the situation that happened later in the war where Charles VII effectively lead the resistance to the English before Joan of Arc came along and helped him become actually crowned king. The thing is, while Jean II was a prisoner the French were largely leaderless. Charles V acted as regent for his father during captivity, but for many reasons the powers of a regent are lesser to that of a king and it was harder to manage a kingdom while the King was away. Had Charles V been made king in his own right by the execution of his father (an execution that would have driven many French nobles back to the French king, after all if the English king is willing to execute a king, what does that say about how he views a Duke or a Count?) he would actually have been in a better position than he was as regent.
Another practical reason not to kill Jean II is that war was fickle, and you never knew when you might be on the losing side of a battle. If you make a habit of executing kings, you can’t expect people to be merciful if they capture you next time lady luck isn’t with you. To some extent the ransom system was an agreement between nobles, a sort of “I won’t kill you, if you don’t kill me” that kept warfare nice and civilized…sort of.
The last reason is that Jean II was worth a lot of money alive, and almost none dead. The term “king’s ransom” didn’t’ come from nowhere, a captured monarch commanded a high price. Waging war was extremely expensive, and medieval kings didn’t have anything like the modern system of taxation that we’re used to now. While the English king actually had a relatively advanced tax system – at least compared to his French counterpart – he still collected taxes intermittently by agreement with the nobles (and increasingly the town burghers via the House of Commons) in Parliament. Given the costs involved in assembling an army, equipping and supplying that army, and transporting it across the Channel to France, war was expensive. Much of these costs could be recouped via successful raiding campaigns and victory in battle, but that money came back in the form of ransoms. By ransoming Jean II, Edward III could refill his coffers for future military campaigns while bankrupting his opponent – making it harder for them to refill their coffers. By forcing the regent, Charles V, to pay his father’s ransom Edward was preventing Charles from being able to field an effective army, one that could prevent the chaos that followed Poitiers and resist future English campaigns and demands.
Overall, had Jean II actually died in the Battle of Poitiers it probably would have put his son, Charles V, in a better position for resisting the English than his father’s capture and ransom did.