The speedometer on my bike broke, and I don't know what information I would need to replace it
The speedo cable turns at a rate proportional to the speed of the wheel or gear that drives the cable. That speed is determined by what's been specified by the engineer who designed the entire system.
In more than 40 years of motorcycling and mechanics, I've never seen the speedo cable rotational speed mentioned in specifications for any machine. I conclude that the answer to "How fast does the cable turn?" is "It depends on the manufacturer and the model." There is no "standard" speed.
When buying a replacement speedo, one specifies the motorcycle's make and year and model, not the speed or rate of the cable's rotation. However, some models do require the buyer to specify a ratio number so that the replacement speedometer will be geared correctly. The ratio number typically appears on the face of the instrument, e.g., "2.6" or some such. I don't know if the number represents the ratio between wheel and cable rotations...but a prudent buyer will be prepared with the prior instrument's number (if one appears) when buying a replacement.