Since the runout on my cheapo rotary tool was too excessive, I set out to build my own spindle.
A closeup of how I built the collet into the spindle shaft. You can see some residual brass from the nut that tightens down on the taper to close the collet. The hole in the spindle body lines up with a hole in the shaft so I can lock the shaft when I need to tighten something down. To make the collet, center drill the shaft, thread the end, then machine the taper. Finally, cut the slit with a nice sharp hacksaw blade. The center hole was the hard part. The Harbor Freight 7x10 just wouldn't give me an accurate enough hole, so I bolted the spindle to my bench, powered it up with a hand drill, then fed a brand new 1/8" cobalt bit spinning the opposite direction in my rotary tool into the shaft. The bit centered exactly and I was able to drill the hole.
A shot of the brass nut that closes the collet. This setup never requires more than hand tightening the nut.
The completed spindle. I built this thing before I had the big red mill, so all of the "milling" to make the flat sections was done with a sander. The drive belt is an O-ring and the motor is a cheap surplus one. This motor lasted about 10 hours before the bushing gave out. I am going to replace this with a nice ball bearing unit soon.
Here are the main components. The shaft pictured was one of 4 that I made before I was able to get one with the hole exactly in the center.
I changed the final design a bit from this picture. I now use threads to keep the whole assembly together.