Unless it is a really old style washing machine then a washing machine motor would l be a nightmare to wire up. They usually have multiple windings so they can run at the different speeds required. Some have variable frequency drives to alter the speed.
A "normal" AC motor such as the ones on "home" compressors, drilling machines etc. are usually single phase induction motors. There are different variants but generally they are single speed - related to the mains frequency of 50 Hz. Two pole motors run at around 3000 rpm and four pole at 1500 rpm. Some can be driven by a variable frequency "power pack" which allows for the speed to be varied within limits. They are of course available in a range of power from a fraction of a HP up to a max of around 4HP (3kW) on single phase electrics - the starting current is usually the limiting factor. They are not particularly cheap.
I suspect it would be nice to have fully variable speed in which case why not look at using an electric drill --they are relatively cheap and run up to around 3000rpm. I would think one with a 600Watt motor would do. (that must equate to around 1/2 HP after losses). It should be fairly easy to make a clamp to hold it in place and they have variable speed built in. Also the chuck should make it easy to grip a drive adaptor to couple to the gearbox input shaft or what ever you wish to test.
You could also look at smaller 12V DC motors used on cars, thinking window winder motors, but it depends on what speed you want and whether they would be powerful enough, and you would have the issue of speed control. Or use a car starter motor - trouble is that won't take any prisoners if there is a problem with the gearbox