How hard actually was it to kill a knight wearing typical 11th-12th Century armour? What were some of the ways in which they typically died in battle?

First off, do you mean 11th and 12th century or the 1100's and 1200's? I ask since Marshall was born in 1150 or so and thus was fighting well over a hundred years after Hastings. I will try to address the period of his liftime with a few notes about the 11th century thrown in.

I am going to focus on the physics of this, and include some general notes about the social history of combat. I hope that u/Rittermeister can provide some more context around how combat occurred in high medieval Europe.

So, first of all, what is mail? Mail is a series of small (several mm, give or take) rings linked together. Most commonly in a '4 in one' pattern where each ring is linked to 4 others. Importantly, European mail of the Middle Ages is not simply 'butted' closed with pliers - it is rivetted shut, or is made of solid links. This means that while the links can be forced apart by piercing blows, it's not nearly as simple as some tests performed with innacurate mail make it seem. This also passes the 'smell test' of common sense - elite warriors of the early Middle Ages would probably not wear armour that could be easily pierced by the most common early medieval weapon, the spear! This isn't to say that mail has no weaknesses to being pierced, just that this is often exaggerated.

Now, mail was worn with some kind of padding under it, at least some of the time. There are references to padded garments called variously aketons and gambesons that may have been worn as independent armour, or might be worn under the mail. When worn under the mail, though, these wouldn't necessarily be very thick - images of mail-clad warriors are uniformly very thin and show the mail is quite form-fitting - this shows up across different media (sculpture, illuminated manuscripts, funerary brasses). And while our surviving mail garments are mostly late medieval, (except the hauberk attributed to Wenceslaus) they don't leave a lot of room for a very thick padded garment underneath. At least in the later middle ages, the really thick cloth armours are worn as armours in their own, right, or are worn -over- mail armour (and in the 14th century, sometimes over plate armour as well). This means that while padding would be present, it wouldn't necessarily be very thick. This means that while the padding is important for making the armour comfortable, it isn't necessarily enough to say, protect against blunt trauma. But this is an area where there is some uncertainty since we have no surviving arming garments of the 12th or 13th centuries.

Add this together and what you have is armour that, as your book correctly observes, is very effective. Allan Williams performed some tests with reconstructed mail to show just how effective it was. He tested a fragment of 15th century mail against a variety of simulated weapons. In this test, he used padding made of 16 layers of linen - this may be thicker than is realistic (see above) but that is hard to tell. He found that mail was defeated (mail and padding penetrated) by a simulated blade with an energy of 170 joules, a simulated lance with an energy of 140 joules and a simulated arrow with an energy of 120 joules. Now, these energies as well beyond what a sword blow, or a sword thrust, can produce. Knight's swords are not armour-penetrating weapons - at most they are designed to go around armour, not through it. I have no firm data on the energy of early 'danish axes' or early polearms, but I suspect that 170 joules is around the upper limit of what one could produce - possible, but it's up there. As for the lance-strike, based on what I've read these energies are possible with a couched lance in a charge (Tobias Capwell in unpublished research has measured 300 joules, though that was using a lance rest). And the arrow impacts are around the energy of a later medieval English longbow* or a crossbow (crossbow energies are not as well researched) - but at least for the longbow are also at the upper limit - keep in mind that the massed deployment of archers was in the future at this point, and it was the sheer volume of fire that made the English so effective in the later Middle Ages - you wouldn't have an Agincourt-like arrowstorm in the 12th century.

So what does this tell us? A knight's mail armour was nearly sword-proof. It was somewhat lance resistant, but could be penetrated. Blows from two-handed axes, early polearms (which became more common in the 13th century, after Marshall's lifetime) or crossbows that struck it in the right conditions could penetrate, but in less than ideal circumstances (bad angles, longer ranges, etc) might not defeat the mail. Knights would be vulnerable to broken limbs from the strongest attacks, but also vulnerable to the fatiguing effect of repeated bruising blows.

But just telling a story of armour penetration is only a fraction of the story - because the best way to defeat armour is to go around it, not through it. Knights in Marshall's day were only beginning to wear helmets that covered their faces, so the face often would be vulnerable. Full mail chausses (mail leggings) were not universally worn in the 12th century and were unknown in the 11th - indeed many if not most 11th century knights seem to have worn no armour on their legs below the skirts of their hauberks (see the Bayeaux Tapestry). Blows that went under the skirt of the hauberk and over the martial chausse could cause a thigh or groin wound.

In the end a man in armour, on his feet and fighting, is very hard to kill, though - it is hard to get around his armour, and harder to go through it. But the story is different when you are able to force him to the ground, strip him of his helmet, and either capture him for ransom or kill him outright. And that is the key - much of the killing happened in the route when one side fled because they lost their nerve, were taking slightly worse casualties, panicked because of a sudden change in the battle, or were simply exhausted and overwhelmed. Isolated men, outside of their formation, could be overwhelmed and picked off. If you were capturing and ransoming your opponents, you would also pick up most of these in the route. Pre-modern battles are not simple slaughter-fests, but struggles of morale and cohesion - and all the armour in the world will not save you if that breaks down.

Sources:

Alan Williams - The Knight and the Blast Furnace

Strickland and Hardy - The Great Warbow

Edge and Paddock - Arms and Armour of the Medieval Knight

*It is hard to say just how strong bows were in the 12th century. We have reports of the Welsh using them very effectively against mail-armoured men in the end of the 13th. But it is quite possible that 12th century bows were of a significantly lighter draw weight than later medieval bows.

Although mail armor ("chain mail") was not foolproof against strong sword cuts, a fighter in full plate armor was however effectively immune to the edged blows of swords. There are no real-life accounts of edge blows effectively cutting through an armored harness; that is one reason why plate armor was so popular and so much effort put into perfecting it.

Though swords were not capable of cutting through plate armor, a fighter would not avoid striking edge blows against an armored opponent if it might bruise or stun him, knock him about, tear into or crack open his helmet or visor, slice through straps and tear off pieces, or otherwise weaken his defense against a more effective technique such as a thrust. While sword cuts that would have been debilitating or lethal on bare flesh might have no effect against soft or hard types of armor, if delivered with great force they could sometimes traumatized the tissue and bone beneath and thereby incapacitate a target. Although, to be accurate, not all armor was of equal quality and some type of helms could indeed be partially split by edge blows from swords. While there are many images from Medieval sources of swords cutting into armor or through helmets, nothing in the historical accounts of actual armored combat or the voluminous instructional texts on armored fighting supports this as being common.

Modern experiments, when performed under realistic conditions with historically accurate weapons using proper technique against historically accurate reproduction armor, have yet to convincingly duplicate what is depicted in such images. An armored fighter was still vulnerable to sharply-pointed tapering swords and other weapons employed in thrusting as well as to crushing from specialized anti-armor weapons. Yet even thrusts against plate armor were difficult to succeed with because it was intentionally designed to deflect and resist them, thus gaps and joints were typically targeted. Yet descriptions of fights with specialized weapons designed for fighting plate armor, such as pole-axes and maces, reveal even they were able to pierce through armor only infrequently. More often they were effective in simply denting and cracking armor to stun and bruise the wearer into a vulnerable condition. But, given strong effort and a hit to the right spot, a rigid point stabbing strongly could puncture armor even if its cutting edge would not.