What happened to the Sorting Hat after the Battle of Hogwarts?

Question

In Harry Potter and the Deathly Hallows our dear friend Voldy sets fire to the Sorting Hat while it is on Neville's head, yet Neville manages to escape and pull Gryffindor's sword out of the burning hat, and the story goes on. From this moment, the Hat completely disappears from the scene and is forgotten by everybody, left alone burning on the ground.

Did anybody put out the fire that was burning the Hat? What happened to it after Big V's death? Was it repaired or changed ?

EDIT POST-ANSWER

As alexwlchan pointed out, the Sorting Hat is clearly still in use 19 years later, so I'll assume it has been saved. Is there any canon information about who repaired/extinguished the fire of the Hat?

By canon, I mean the books or JKR interviews.

Answer 1

Both Neville and the Sorting hat appear to have been protected by the "aura of love" cast over them by Harry Potter.

Although they're both engulfed in Voldemort's magical flames, there seems to be little or no consequence of this and the flames themselves seem to disappear as soon as Voldemort is distracted. I've edited for brevity and clarity:

‘Neville here is now going to demonstrate what happens to anyone foolish enough to continue to oppose me,’ said Voldemort, and with a flick of his wand, he caused the Sorting Hat to burst into flames.

Screams split the dawn, and Neville was aflame, rooted to the spot, unable to move, and Harry could not bear it: he must act –

In one swift, fluid motion Neville broke free of the Body-Bind Curse upon him; the flaming Hat fell off him and he drew from its depths something silver, with a glittering, rubied handle –

_{^{"Harry Potter and the Deathly Hallows"}}

Later, Harry explains to Voldemort why his spells aren't working very well...

‘– I meant to, and that’s what did it. I’ve done what my mother did. They’re protected from you. Haven’t you noticed how none of the spells you put on them are binding? You can’t torture them. You can’t touch them. You don’t learn from your mistakes, Riddle, do you?’

_{^{"Harry Potter and the Deathly Hallows"}}

Even if there was any burn damage after the magical flames die down, I presume that fixing the hat afterwards would be trivially easy using a "mending charm" or some similar magic spell.

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As to the ultimate fate of the hat, as @alexwlchan notes, Harry Potter indicates that they still use a sorting hat in 2017. His dialogue strongly suggests that it's the same hat that sorted him into Gryffindor:

”The Sorting Hat takes your choice into account.”

“Really?”

“It did for me,” said Harry.

_{^{"Harry Potter and the Deathly Hallows"}}

Answer 2

The hat was not destroyed beyond repair at the end of Deathly Hallows, as we learn in the epilogue that it is still being used to sort students when Harry’s children are at Hogwarts:

”The Sorting Hat takes your choice into account.”

“Really?”

“It did for me,” said Harry.

However, I don’t think the books explain how it was rescued.

Answer 3

It is kind of confusing, but after Voldemort set the sorting hat on fire (along with Neville's head) both Neville and the sorting were protected by the power of love.