Hmmm - Interesting hypothesis in this link, but probably nonesense. Henry became king 10 years after the battle and was known to have done much carousing in the interim. His behaviour as king is all to do with careful image management combined with both piety and a desparate need to legitimise his hold of the crown.
First, 'penetrating head trauma' doesn't necessarily, or even particularly often, mean penetrating brain injury, on account of that organ being well protected inside a box with thick bone over vulnerable parts and thinner bone where lots of other protection exists. Second, the temporal lobe, as illustrated in the link, lies in the middle cranial fossa which creates a bulge to about half way down the orbit. In order to stick an arrow in it according to the type of injury being discussed ie a frontal strike, you either need to lose your eye, or if hit in the cheek, angle it upwards by at least 30 degrees (unlikely given that the arrow came from above and Henry was fighting up a hill at the time). In any case, it isn't six inches back. Third, if there was any penetration of the middle fossa resulting in a brain injury which could lead to the sort of stuff mentioned, there would have been a CSF leak (brain fluid). There are particular features of that kind of trauma that in this context, meningitis and death would have been pretty much certain no matter what anyone did. Finally, the lack of evidence of loss of consiousness goes against any direct brain injury.
I have thought about this injury often and have heard a couple of comments relating to the arrow being embedded in the skull or even the mandible(!). My own take on the descriptions is that it entered the cheek beside the nose probably along a more or less true horizontal line (Henry would still have had to have been looking up for an arrow to come down and strike in this way), went through the maxillary sinus and across the infratemporal fossa(below the middle fossa) losing energy as it did so and got itself stuck in the undersurface of petrous temporal bone , which is a long way back and certainly could be assessed as 6 inches if the cheek was swollen as it would have been. The petrous temporal bone is, as the name suggests very hard -in fact the hardest bone in the body and winkling an arrow out of it would certainly be tricky. The internal carotid artery would have traversed very near to the path of the arrow so survival was very fortunate indeed - interestingly, had the arrow hit beside the nose and gone up into the temporal lobe, it would probably have severed the internal carotid on its way!
Hats off to Bradmore - his actions give the lie to the idea that all medics of the time were a waste of space - although it has to be said that he was known to be unconventional and was under something of a cloud at the time he was called in.
C