harry potter - Why didn't Dumbledore ever heal his nose?

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!