. . . I think that I will attach the new mirror, whether first or second surface, with the double side tape.
Many double-stick tapes use an adhesive that won't stand up to heat. Typical ratings are for a max of 150F or 200F -- and old tungsten or halogen units will often get hotter than that.
You might want to experiment with your chosen tape and a heat gun first -- or look up the adhesive if the manufacturer provides a spec. Alternatively, McMaster Carr and others sell higher temperature double stick tapes (variously good to 250F, 300F, top around 500F).
Thanks PeteM for this important comment.
Since the old mirror was defective, I already replaced it with a new second-surface mirror (at least until I get the first-surface and try it), using a No-Name double sided tape. I had been concerned about heat from the lamp, so before I installed it, I checked with a Fluke digital thermometer and a bare-bead thermocouple sensor. Placed the sensor tip inside the assembled (without mirror) field aperture unit, near the back. Turned on the 10W LED illuminator (unfortunately, no halogen...). Room temperature was 75F. The temperature inside the frame rose by about 18F, then leveled off. This is a rough measurement for sure, heat dissipation varies, and the scope stand was lying on its side, so heat losses by natural ventilation were optimal.
Heat dissipation mainly occurs via conduction through the aluminum base of the stand. I believe, that even on a very warm day (say 90F room temperature, no AC), the mirror temperature will not reach 120. Still, I will keep that in mind.
P.S. One advantage of this commercial LED illumination setup is that the multi-fin heat sink of the lamp "housing" is outside the base of the microscope, not within the space under the stand as in some home-built LED conversions. Hence, much or most of the heat created by the LED supply is dissipated outside the scope itself.