Step 1: Take apart an old cordless drill
Step 2: Mounting the drill
I recommend starting off by drawing a centerline on your base to assist with locating the rest of the mounting points (something I didn't do).
I carefully glued and screwed the motor mount block to my base. and then mounted the motor with a short length of pipe-strapping tape and a couple screws on each side. This actually secured the motor quite well, but I later realized that the unsupported plastic gearbox assembly gave my drill bit at the end of the chuck way too much play. The bracket seen on the right gets added at a later step to address that problem.
Step 3: The sliding stage
The simplest way to make sure that everything ends up being aligned correctly is to basically machine the hole for the dowel in-place on the slide. I started with a 1/4" Forstener bit, which matched my dowels, and drilled a hole to the depth of the dowel length simply using my hand to push the stage/jig against the drill.
With the primary hole drilled, I removed the stage from the slide and moved over to a drill press where I used a 5/8" Forstener bit to drill a hole carefully located just above the first hole so that it intersected the first hole just above the lower hemisphere. I then used a saw to cut away the front top half of the block to reveal the carefully machined trough to hold the dowels and provide access for insertion/removal.
Step 4: The handle and linkage
A piece of aluminum scrap from someone practicing a T-weld provided a bracket for the back of the stage to attach the actuating linkage, and also another bracket to stabilize the drill bit and possibly provide any fine adjustment needed. I simply held the bracket by hand, against the base, and then pressed it against the bit to drill a hole at the proper height. A couple small L brackets helped attach this to the base and aren't really visible in the pictures.
I little eyeball guess work and trial and error quickly derived the ideal linkage length and control arm mount position. I drilled a couple holes in the lever and mounted the trigger assembly from the original drill using a couple zip ties. This was wired up with some scrap to the motor and a terminal for connecting to a small 12v battery. A little bit of foam pipe insulation and duct tape finished up the lever with a comfortable handle. The MOSFET and heat sink for the speed control can be seen dangling a bit from the handle in the picture.
Step 5: Put it to use
A slightly more advanced version could feature a second linkage mounted higher on the control handle with a spring system that could be used to automatically hold the dowel and allow for single handed operation.