How can Batman run and climb in the pits without his knee brace?

I don't remember seeing Bruce Wayne without trousers or with short trousers in the pit.

He could still wear his knee brace, but why Bane would let him keep that would be a another mystery.

I think the knee brace was originally just to allow Bruce to begin walking again.

There are many ways knee-cartilage can be replaced as seen in this article.

So Bruce could easily have had one of these procedures to fix his knee during the time between using the knee-brace originally and being thrown in the pit which would explain why he doesn't need the knee-brace in the pit.

The whole cripple thing might just be a ruse from Bruce Wayne.
Thanks to that ruse, people let him alone when there's a party at the manor in the beginning (a bit like when he pretends to be drink in the previous movie).
Then thanks to that ruse again, he's able to secretly contact Gordon at the hospital, pretending a visit to the doctor. A minute after hearing the diagnosis, he jumps down several floors to chat with his friend.
This is probably painful anyway, but not as much for him as he'll tell people. As mentioned before, he's the Batman ! And can overcome by his master mind and iron will what his body would seem to refuse doing.

The inspiration for The Pit in the movie is (or can be found - Nolan, as far as I know, never said anything about it) found in Lazarus Pit from DC Universe (comic). Lazarus Pit is very well known for it's mystic healing abilities.

While this jumping and climbing from the point of view of Dark Knight Rises might be indeed plot hole, it is actually very plausible if we look at the whole Batman universe.

He was in the pit for 5 months. That's plenty of time to get fit. Though I still don't believe it's realistic, considering all his injuries. I think the broken back thing is just staying true to the comics and the will power of Batman.

IMHO, I think Bruce Wayne was still wearing his knee brace. We never saw his knee to confirm that Bane took it off.

We just assumed that Bane took it off, because it seems logical. However, think about it from Bane's perspective:

Bane was talking about breaking Batman. He said that he would break his body and soul. He did break his back leaving him crippled. All was left is to break his soul.

He was so sure that Bruce wouldn't even try to fight back or come after him (Remember his surprise when he saw him again).
A man so sure of his victory won't bother by simple details as a knee brace (which is neither a weapon or a communication device) when your opponent is already half dead and prisoned in a pit he himself failed to escape (again remember when he was fighting batman and bragged about being born in the shadows contrarily to batman who training in shadows ).