You can improve the reel's grip on the shaft by grinding a flat part into the shaft. However, the set screw is in a plastic piece that can very easily break, so over-tightening can do serious damage.

If the reel is spinning in place when it starts up, I don't think being loose on the shaft is your problem.

There's lots of info around about those damn slot reels. Here's something I referenced in the past: https://keepingmycool.blogspot.com/2013/04/how-to-clean-who-dunnit-slot-reel-motor.html

According to the manual, the set screw is part 4006-01076-04 (Set Screw 6-32 x 1/4"), so...

An allen wrench. A small one. This page says it's 1/16". It's definitely in inches (not metric).

Be very careful not to over-tighten the screw!

As some further 'ideas' on this issue, here is how I fixed the admittedly poor design of the Who Dunnit reel set screw. The original design is poor because the shaft does not have a flat, as noted, but also because the set screw is far stronger than the plastic reel and can easily crack the plastic. With the little hub I made out of aluminum, the set screw can be tightened fully without issue. The hub also helps stiffen the reel...on this machine the reels had a good bit of axial runout (side-to-side wobble) and even though the hub is not very large in diameter, it made a good difference in reducing the wobble.

The hub is 1.2" diameter with (4) 6-32 screws on a 1" BC. The center is 5MM, and the overall height is 1/2". The small diameter of the hub is .687" and the backing plate is also 1.2" diameter and 1/8" thick with a .377" center hole.

The photo shows the original set screw layout and the hub installed, Another uninstalled hub/backing plate is shown on the table.

Who Dunnit Reel (resized).jpg

Those look awsome wish i could get a set
Keep up the good work and keep spinning those wheels.