Doesn't have to reflect the siege of Jerusalem itself, but any siege that could have happened at the time. I've found many questions and answers on the movie, but none on the siege itself.
Many thanks!
Here are some videos of the siege scenes in Kingdom of Heaven for experts to look over.
As with many aspects of Kingdom of Heaven, it has a broad sweep of truth to it but it falls apart a bit at the details. In particular, the depiction of the Siege of Jerusalem is quite flawed but it does contain elements that would have been common in contemporary sieges.
One thing that's somewhat frustrating for this subject is how little Crusading scholars seem to comment on the Siege of Jerusalem. It is by and large treated as an inevitable afterthought in the wake of the Battle of Hattin. To some extent this appears to be related to the sources we have on it. There is a lot of information, and historical discussion, on the actual surrender and how it was implemented but not much on the siege that preceded it. As an example, Jim Bradbury's The Medieval Siege skips it entirely, moving from the Crusader siege of Damascus to the Siege of Acre with no mention of the period between the two Crusades. It's very possible that R. Rogers' Latin Siege Warfare in the Twelfth Century and Peter Purton's A History of Early Medieval Sieges (which includes the Crusades as Purton treats everything pre-1200 as Early medieval) say more on the subject but I, unfortunately, do not have them at my fingertips. What I do have is Thomas Asbridge's The Crusade: The War for the Holy Land and Ann Marie-Edde's Saladin. Well...and the aforementioned unhelpful Bradbury book...
One of the main reasons the Siege of Jerusalem receives so little attention is that it's result was almost inevitable. Balian of Ibelin (who himself had to receive permission from Saladin to even stay and defend the city) had nowhere near the garrison size he needed to effectively defend the city from Saladin's vast army. The city had a lot of people in it, that's for sure, but very few combatants. He did actually take the rather unusual step of knighting vast numbers of the populace (although not quite so many as in the movie) to assist in the defense but even then it was going to be hard. Since Saladin had crushed the main Crusading army at Hattin there was no real hope of a relief force coming to save Balian, either.
The broad sweep of the narrative is correct, though. Saladin assaulted the western city walls, and was rebuffed, and then moved his camp to the north east of the city where his siege weapons opened a breach in the wall. The sources I have are vague on whether he actually tried to assault the breach or not, but given his initial demands that he wanted to slaughter the entire city like they did in 1099 it seems likely. Anyway, the breach was not taken by Saladin and it forced Balian to come to terms with Saladin, which in the end were quite generous.
The biggest problem with the Kingdom of Heaven depiction is honestly the scale, particularly with regard to the number of siege engines. Saladin is shown as having at least 6 siege towers, which is far too many. The Crusaders took the city in 1099 with only two while the Siege of Acre in 1190 employed only 3 (unsuccessfully I might add). The thing is, building siege towers was extremely time consuming and Saladin did not spend that much time outside of Jerusalem before he took it. While he certainly moved siege weaponry with his army, likely by disassembling it, it is more likely that it was trebuchet's rather than siege towers. Siege towers are also primarily a Crusader technique and I'm not aware of any instances when Saladin employed them, that said he did fight a lot of sieges in his career so I could have missed one.
The ballistae that were counter-weighted to topple siege towers are almost certainly not a real thing. Standard practice for destroying siege towers was to set them on fire, ideally with Greek Fire, which is what happened at Acre.
Siege ladders were definitely a thing and part of almost any assault on a fortified position, nothing really wrong there.
The counterweight trebuchets are reasonably accurate, although again there may be too many of them. It's hard to know exactly how many Saladin had in his army, though, since we don't have exact records. The counterweight trebuchet is generally dated to around this period and there is a convincing reference to one in a military treatise composed for Saladin during his lifetime. That said, we don't have conclusive proof that he was using them at this particular point in time on this campaign so there is an element of doubt there. He could have been using more traditional catapults.
Since Saladin did not camp outside Jerusalem very long, his army arrived outside the city on the 20th of September and the city had fallen by the 2nd of October, there was not much time to build siege engines. This means that he must have brought his trebuchet's with him from his previous' sieges, no small feat it should be mentioned, and that suggests he would not have had a vast number of them. He had just finished his siege of Ascalon on the 5th of September so he only had 15 days to get to Jerusalem with his army (and on the way he negotiated the surrender of Gaza, Darum, Ramla, Yubna and Latrun, so it wasn't exactly a rushed march) and it seems unlikely he could have made that march in that time with the number of trebuchet's shown in the clip (much less the enormous siege towers!).
The breach was opened by Saladin's siege weapons pretty much like the movie shows, again it's not clear if he assaulted the breach but it seems likely, and that breach forced the immediate negotiation of terms of surrender.
The negotiation of the terms of surrender shown are, perhaps, overly generous to Saladin. What we do know is that Balian threatened to destroy the holy sites of Islam (although no mention of him destroying Christian ones) as well as butchering all the Muslims currently in the city (quite a few, estimated at 3-5k by Edde). What is also suggested, though, is that Saladin declared he wanted to butcher the Christians in the city in payment for 1099 before Balian issued his threat. How committed Saladin was to this has been hotly debated. Some historians have argued that he was simply saying this to appease the hard liners in his army and that he did not reasonably expect to follow through. This is based on how peaceful he had been in previous surrender negotiations with Christians as well as a plethora of contemporary literature arguing that Saladin should be more brutal in pursuing his Jihad. Saladin also didn't just offer to let everyone go, there was a matter of ransom. The entire city was ransomed and while the price was quite low quite a lot of Christians couldn't afford it and were enslaved as a result. Balian paid 30,000 dinars to ransom destitute Christians and, reportedly, Saladin and his brother redeemed and freed quite a few as well. Still, between 15 and 16 thousand Christians were likely enslaved. Somewhat scandalously, the Patriarch of Jerusalem reportedly fled with all the church's money and didn't pay the ransom for any of the enslaved Christians. Eastern Christians (mostly Armenians but some Copts) had to pay the ransom but if they did so they were permitted to live in Jersualem and retain their property.
I understand why the whole ransom bit was cut from the movie, though. It was remarkably generous for its time but to a modern viewer could appear cruel and Scott clearly wanted to show how generous Saladin was in his terms of surrender without having to explain the contemporary practices around surrendering in the middle ages.
Edit: Sources in the text but for ease of reference
Thomas Asbridge, The Crusades: The War for the Holy Land
Anne-Marie Edde, Saladin
I'm not sure which siege of Jerusalem was the one in the movie (never seen it).
The siege of Jerusalem in 1099 however (the one where the Crusaders took the city in the First Crusade) had some fairly major differences from the clips shown.
It was a fairly short siege. The Crusaders reached Jerusalem on June 7 and took the city on July 15, 1099.
The Crusaders needed it to be a short siege, because the city was well provisioned, whereas the Crusaders had very few supplies of food, and the defenders had implemented a scorched earth policy in the surrounding countryside, so there was little food to be found around the city.
When they reached the city, the Crusaders spent 5 days searching for wood (the defenders had burnt or removed all the wood close to the city) and building scaling ladders. On June 15, they assaulted the walls with ladders. This attack was repulsed by the defenders.
Luckily for the Crusaders, the day after this defeat, two Genoese galleys sailed into the port of Jaffa. Not only did these ships carry the supplies that kept the Crusaders from starving, but the ships were broken up to supply the timbers to build two siege towers. With the wood from the galleys, and some other wood they found foraging, the Crusaders had enough wood to build two 50 foot siege towers , one battering ram, and several catapults.
By July 14th, the galleys had been broken up, the wood had been transported up from the coast to Jerusalem, the siege engines had been completed (they must have worked like fiends to do all this in the time), and that night the Crusaders mounted their attack.
One siege tower advanced on Jerusalem from the south, the other from the northwest.
The defenders destroyed the siege tower to the south with fire before it reached the walls. In the north west, the tower took two hours to advance to the wall but it reached the wall and the Crusaders stormed across, took the wall, and eventually opened a gate so the other Crusaders outside could storm in.
The main differences from the movie clips:
The attempt to storm the walls with ladders was actually an entirely different attack from the one with siege towers, whereas the movie showed them as part of the same assault.
The movie showed multiple trebuchets slinging rocks at the city. In fact, the Crusaders may not have had any trebuchets, and only a few catapults.
The movie showed many siege towers. In fact the Crusaders only had two, and only one of them managed to reach the walls, but that was enough.