I do not think that the speed of a single phase motor is variable. You will need to change your motor to a three phase one. Once you do this you can get several makes of single phase to three phase inverters to control it. Mine is an Omron (quite old) but very servicable. I changed my motor from a 1hp single to a 1 3/4 hp three phase motor - and controlled it with the Omron manually. The Omron gives full speed and direction control. The torque of the motor falls off if you try and use it below about 50% speed (30 hz), so I will be putting in some gearbox/pulley arrangements to turn some of my larger pieces.
Unlike Hood, who uses a slot detector, I use a reflective detector - same principle - ir.led shining and reflects onto a detector - mounted 5mm from my chuck mounting plate (painted matt black). A small reflector, made from my granddaughters "sticky" bits and pieces, stuck to the edge of the mounting plate, reflects the IR back once per rev.
As I put in the original post, it works well, but seems to over read. I put this down to a 12 volt supply to the Omron instead of 10. I will work on that.
By the way - one inverter (providing it is large enough for your largest machine, can supply all your machines, you do not need an inverter for them all. (You can only use one at once, though)