The castle has curtain style walls that are thirty feet high, and of a relative thickness that supports them solidly. They are on the flatlands of France, and the Castle is occupied by roughly two hundred soldiers. What kind of siege weapons do I bring, how many men do I bring, and how do I go about successfully conquering the castle? Success can be defined as cessation of armed resistance inside the walls.
There were no sieges of castles in France in 1400. France was enjoying an uncharacteristic lull in the 100 Years War against England. War on a large scale would not resume until 1415, when King Henry V invaded France.
If there had been a siege at this time, however, the besieging forces would have had some tactics which they could use.
Cannon, though primitive ones had been introduced, were not yet powerful enough to be seriously threatening weapons against a castle of this size and scale. (Though they soon would be, as in 1453 cannon were instrumental in the siege of Constantinople.)
The besiegers might try to assault the castle, with ladders and siege towers, with covering fire from archers and trebuchets. This, however, was unlikely to be a successful tactic against such a strong castle, with such high walls and such a large garrison. Simultaneous attacks against the whole perimeter might succeed, but even that would be unlikely, and very costly in casualties.
It might be possible to negotiate with the defenders, and entice them or bribe them to surrender. If that didn't work, it might be possible to find a traitor who could be bribed to open a postern door or let down a rope ladder from some secluded point on the walls.
Barring those solutions, the castle could be besieged. Depending on how well the castle was provisioned (which could probably be discovered) such a large garrison might not be able to hold out for long before starting to run out of food.
If the castle was well provisioned, and it was important to capture it, the only option left was mines. The castle is built on flat land, so presumably not on bedrock. Therefore, a mining operation was possible.
Mining operations took a long time, and were dangerous, but they often worked.
At Caen in 1417, and at Melun in 1420, King Henry V used mining to attack French fortresses. The French defenders used bowls of water to detect English mines by watching for ripples in the water caused by English mining operations vibrating the ground. Then they would dig "counter-mines" to try to disrupt the miners.
King Henry himself is said to have gone down into the mines at Melun to fight against French counter miners.
At the siege of Caen, the English mines opened a breach in the walls through which the Duke of Clarence's men forced an entry, taking the city. The citadel surrendered two weeks later.
At the siege of Melun, the English and Burgundians had 20,000 soldiers versus 700 defenders. Breaches in the wall made by the mines were successfully defended by the French (illustrating the great defensive strength of castles in the era before powerful cannon) but after a 5 month siege, the castle was forced to surrender due to lack of supplies.
Admittedly I am outside of my area of expertise as it is medieval France, but I think I can answer at least generally drawing from my knowledge of sieges at large.
Depending on how much traffic the castle gets, you would want a minimum number of troops for a variety of reasons. In other words, the fewest amount of troops necessary to complete your objective.
First of all, feeding an army on the move and away from home is expensive, complicated and difficult. It's also a necessity and only gets harder the more soldiers you bring.
Also, many men mean new difficulties in other areas of logistics, particularly orderly travel. Assuming you muster your forces in a staging area and then travel together to the target lands/castle, more soldiers mean slower movement, meaning more supplies and money needed as well as time. This growth in the numbers of an army can grow exponentially because for every fighting soldier, generally there were at least 1, maybe 2 people supporting him in the follower's camp.
An army of 100,000 would have been a core of maybe 20,000 soldiers with many others being common laborers, camp followers, some auxiliaries etc.
So in the case of numbers of troops you would bring, less is more. Of course this changes as circumstances change but in general.
At the same time, the general rule of thumb is that to actively take a defensible position from an enemy, you need more manpower. This is because defensive positions, like a castle, are force multipliers and are inherently made to give an inferior force the chance to hold out against in the face of a superior force.
Of course, this is assuming that the attackers will be trying to take the castle by force of arms.
Unless your siege is exceptional in circumstances or has specific time sensitive goals like "we must take this castle in a week so we aren't crushed by a relief force coming to this castle's aid", generally a siege is less about intense Helm's Deep style siege combat and more about posturing, shows of force.
Everyone knows the first thing the besieging force does is cut off all the roads and control traffic into and out of the area.
The thing is, this is usually the only thing a besieging force has to do.
Now this might seem dismissive of the pressures and seriousness of the difficulty of conquering a castle, but the vast majority of sieges throughout history ended without a pitched battle, with men falling from walls and fiery artillery blowing down gates.
That being said, it's simple to say 'surround the castle, then just wait' but it's something else to actually do so.
You need enough men to control the flow of people and traffic as well as exert some power in rough country, ie areas off the roads or otherwise people will just walk where they please, if with a distinct lack of wagons. You also need enough men to defend against possible sorties from within as well as defend against relief forces coming from outside to break the siege.
Usually, besieging armies outnumber the defenders by at least double so around 400 men ignoring other factors to do everything they need. Triple or quadruple is not uncommon at all but generally only fielded in decisive maneuvers, not for the average siege.
The idea again being that the besiegers would only need enough men to hold the castle's defenders by their throats and let starvation, disease, and despair do their job for them. The most important thing to remember is that for a besieger a pitched battle is incredibly risky and they would avoid having to do so if they could. The best siege was one that ended with no battle.
As for specific siege weapons, for such a small scale castle, one would most likely bring nothing.
Again, while sieges were incredibly common, actual full on siege battles were less so. If siege weapons were involved, usually for use against a much larger fortress, they would commonly be built onsite or nearby where the construction material was available. Rams, rudimentary ballistae and catapults, maybe ladders, pavise, etc.
The main way to successfully conquer a castle in lieu of actual head to head combat would be negotiation skills. If you convince the defenders that they would not be able to hold out longer than your army can and only death by starvation await them, then they would have little choice in the matter.
In fact, in Japan, most sieges ended by civil debate and discussion. Both sides would meet for parley, argue their sides for why they would win handily. If it was agreed that the besiegers would inevitably win, they would take the castle peacefully, no fuss. If the defenders would win, the besiegers would just pack up and go home. In case of disagreement, the siege goes on.
Even in the case of no surrender, sieges were incredibly effective at defeating defenders because of the limited amount of food and supplies a castle could hold, especially if it was providing shelter to a large civilian population. The word siege often evokes this image of years and years of mind numbing waiting and tension, punctuated by minor skirmishes but mostly just waiting.
The reality of warfare was that the length of sieges were often counted in weeks, and occasionally months. Sieges that dragged on for years, a la the siege of Troy, were a rarity anywhere in the world, which is why they are so well remembered and so stark in many people's minds.
So the main thing to take away is, if you were a siege master laying siege to a castle, you would likely not expect very much fighting, if any at all. You would have plenty of experience cordoning off the surrounding lands and roads, know how many patrols you need to keep men from sneaking in through the forests and rivers. You would have been able to bring just enough food to feed all your men if you really stretch it out near the end. And you would expect the castle to fall very routinely, the way all the other sieges ended, and hope that enemy reinforcements don't pounce on your forces that have just enough men to pull this off.
Taking any castle is no easy job and there's tons of pressure to succeed but you'll be fine. This castle is no mighty fortress of thousands strong. And after all, this isn't your first siege.
I was speaking about sieges in general so I hope I wasn't too far off! Hope this helps! Cheers!