I was thinking along the same lines before I reached the end of this thread. I even ordered some acrylic rods this afternoon to cut in half: the semi-cylinders just might work as acceptable condensers. An oblique fs-mirror allows a 90 degree angle in the design (leds vertical) and perhaps fine adjustment of the light rays' inclination.
How much led-power do you need, I wonder? If as much as I think, the strip leds you're using wouldn't be my first choice. They have been made for easy installation, but the diodes are widely spaced and in the wrong orientation, with their long axes vertical. Even then, there are strips at least as narrow as 3mm - did you select one of these extra narrow strips?
If much light is required, I am inclined towards smd leds, a series of the x014 leds or even the 1206 (for those not familiar with these numbers: the four figures denote the dimensions of the led module in 0.1mm, so 7014 is 7.0x1.4mm). Soldering these tiny devils may be hard, but in series it shouldn't be too difficult.
Nice to find the thread revived. There was a discussion about the design, the possibility of directly using tiny narrow and well-oriented LEDs was suggested (by MichaelG). For me, handling and soldering such "glowing microbes" was beyond my capability, so I went the easy road and used the densest LED strip I could find. It is 204 LEDs per meter, and the LEDs are 3014, so they are 3.0mm in height. Certainly, this LED strip is far from ideal, but is easy to work with.
I managed to attach it more or less horizontally on a metal plate carrier, that eliminates all heat problems, and equipped with leveling screws. I masked most of light beam of each LED, except for the center, by means of a black cardboard strip that runs end-to end in front of the LEDs. Shown below as a black strip. So, I could send a relatively narrow beam into the edge of the slide.
Later, I devised a "convex lens": cut a 30mm section of a 0.6mm-diameter quarz optical fiber; exposed it totally, and epoxied it in front of the slit. The right and left ends of the fiber are epoxied (a crude job really). This cylindrical lens is located exactly in front of the unmasked centers of the LEDs.
The reason I chose a quarz fiber rather than acrylic fiber is exactly to minimize light loss.
Nowadays I have a stereoscope so better equipped for such jobs, but I am still hoping for someone else to come with a better gadget of this sort, maybe!
Please note, that it was all made using just plain hand tools and no machines.