Spoiler for New Scientist Enigma 1532:
I was pleased that the method I came up with to solve this puzzle was the same as that of maura lessing, which was easier and more elegant than any of the other posted answers.
But I also was curious what happens as one sweeps the angle of the laser from the initial 0° to the final 9°, so I wrote a program to generate pictures which I combined into an animated GIF. Here it is. The beam enters at the left center corner. You’ll see some abrupt and drastic changes in the path, but if you focus on the initial beam you’ll see it’s moving continuously across a portion of the upper right mirror. The big changes are simply what happens when a reflected beam crosses from one mirror to another. The behavior of the final, escaping beam is rather chaotic.
Of course the length of the path and the number of bounces taken before escape depends on the size of the gaps between the mirrors, except for those angles where the beam intersects a corner exactly.