That doesn't explain how real Dumbledore didn't see it, though.

Dumbledore may have seen it, and refused to let on. Quirrell is next to impossible to kill permanently. He says he can detach from a body if it is in danger, and if you kill the body, he detaches too.

The argument would go that Dumbledore was luring Quirrell into the Mirror as a final solution, and therefore pretended not to recognize him.

(Though wouldn't an easier play be a sneak attack by Dumbledore incapacitating Quirrel but not killing him? )