Looking at the photos, this MAY be an external cavity laser. The
typical HeNe has the mirrors cemented right on the end of the
central glass tube. External cavity lasers have one (or rarely
both) mirrors separate from the laser tube, and generally have
Brewster windows (thin glass plates mounted at about 56 degrees
on the end(s) of the tube.) The external mirror will have
some form of adjusting screws to align the mirror.
I used to be the master mirror aligner at our lab. My technique
was to shoot a working laser through the dead laser's two
mirrors, and observer the spot on a card at the far end. Generally
you could see a 2nd reflection on the card, and try to steer them
together. if the dead laser was being excited, you would occasionally
see short flashes of lasing when you passed through the proper
alignment. As you get closer, you had to make extremely small
adjustments and then take you hands away to let everything stablize
thermally. It is a very frustrating process, but if your laser
is, indeed external cavity, then you may need to do this.
Jon