I have always wondered why Dumbledore didn't have his broken nose "magic-ed" back into place. It is a nice character element with his half-moon spectacles, etc. but one wonders if it is intended to infer that he carried it as a mark of shame and responsibility over the death of his sister, his nose having been broken by his brother at their sister's funeral.
Is there anything more than speculation and inference on this small detail?
I do think this is speculation and inference, the broken nose being a mark of shame, that is. Deathly Hallows tells us:
‘Oh, Aberforth is just the tip of the dungheap,’ laughs Skeeter. ‘No, no, I’m talking about much worse than a brother with a fondness for fiddling about with goats, worse even than the Muggle-maiming father – Dumbledore couldn’t keep either of them quiet, anyway, they were both charged by the Wizengamot. No, it’s the mother and the sister that intrigued me, and a little digging uncovered a positive nest of nastiness – but, as I say, you’ll have to wait for chapters nine to twelve for full details. All I can say now is, it’s no wonder Dumbledore never talked about how his nose got broken.’
Deathly Hallows - page 28 - Bloomsbury - chapter two, In Memoriam
If Dumbledore did or did not fix his nose due to guilt and shame, it sounds like he never told anyone, either way.
J.K. Rowling has said that magic doesn't heal all wounds. Perhaps a nose broken out of anger can't be healed by magic, especially if it's righteous anger. For example, look at Mad-Eye Moody -- he was covered in scars from injuries, and lost both an eye and a leg, and there is no indication in the books that Moody chose these injuries instead of having them magically healed. Canon suggests that Moody's wounds could not be healed; Molly Weasley notes in Deathly Hallows that injuries caused by Dark Magic cannot be magically healed, which is why George had to go without his ear.
As J.K. Rowling has said about Snape and his poor hygiene and bad looks, perhaps, regarding Dumbledore's slightly disfigured nose, Dumbledore valued other qualities in himself than his looks and therefore never had his nose fixed.
However, according to Rita Skeeter, we will never know for sure!
He may have decided to leave his nose crooked as a daily reminder of what his desire for power had cost, to keep himself from being tempted in the future. His broken nose was given to him at Arianna's funeral by Aberforth. Dumbledore felt personally responsible for her death, and it was a turning point in his life. For the rest of his life he was wary of being tempted by power and returning to his old ways.
I, meanwhile, was offered the post of Minister of Magic, not once, but several times. Naturally, I refused. I had learned that I was not to be trusted with power.
...
I was safer at Hogwarts. I think I was a good teacher
Dumbledore has also said before that remnants of injuries can be helpful or serve a purpose, which is why he wouldn't have healed Harry's scar even if he had the power.
Scars can come in handy. I have one myself above my left knee that is a perfect map of the London Underground
