So, basically, the wooden piece with the hooks is fixed to my impact drill in locked mode, while the gearbox is spun with an impact driver (the gearbox is not fixed, because the driver only snaps on hex bits; I'm thinking about gluing the round Technic piece into a socket). Once the plies are twisted, I unlock the drill and spin the hook piece to wrap them all together.
The problem now is that there's a lot of friction in the mechanism. My first version was better in this regard, but one of the plies spun on the main axis, which meant it stayed stationary when came time to wrap the plies, and that messed up the length of each plies.
The only reason it's not 4-ply is because I ran out of gears.