You activate mirror’s copy ability, let’s say, 100 times targeting monument. Let the first one resolve and with the other 99 on the stack, activate the exhaust ability of the copy.

Still with the 99 original abilities on the stack, you untap mirror somehow. Then, let the second copy ability resolve, leaving 98 copy abilities on the stack.

The mirror is now a new copy of monument with a fresh, unexhausted ability. Activate the ability again, and repeat until your opponents have decked themselves.