Instead of modifying the microscope, I decided to use a spare bulb to create a direct replacement. This would allow to go back to original bulbs, just in case. So I took a spare lamp, which is a common bulb mounted in a plate to precisely locate it in the correct position. I removed the bulb and soldered two big washers, one on each side. A threaded rod would allow to adjust the position of the LED in the Z axis while the oversized washers would allow to adjust the position in the X and Y axes. At the end of the rod I placed a small heatsink with the LED and also the switching driver for it. I mixed heatsink compound with epoxy so that heat is transfered from the LED plate to the whole assembly.:

Thanks to the oversized washers and the rod, I was able to adjust the LED in position, which is quite critical. Moving a few milimeters the LED up/down or to the sides resulted in a very uneven illumination of the subjects. This is how the original and LED lamps look when installed in the lamp housing. You can see the nut at the end of the rod, which needs to be loosened to adjust the placement of the LED on the three axes, which by the way should never need to be touched again:

After carefully adjusting the position, the new lamp evenly lights the whole viewing area, is brighter than the old lamp and there is no noticeable heat. This is what the die in a PIC 12C508A looks as viewed from the microscope, much better than needed to comfortably work with the smallest SMD parts. Notice the shallow depth of field even when photographed at f/22. The front lens was about 20cm (8 inches) appart from the IC, which is enough room to work comfortably.