Robb, welcome to the forum. I'm not sure how the 2-speeds are wired or even whether it's a capacitor 'start' or capacitor 'run' configuration. But, the caps are one of the two common failure components. An easy way to test them is with a regular, cheapo, V-O-M- (Volt-Ohm-Milliamp) meter. Less than $20 at Radio Shack. You need to disconnect the cap and discharge it by shorting the terminals together for a little bit. 10 seconds is sufficient. Connect the VOM in the Ohms mode one way and you should see the resistance drop initially then ramp up to some high value. Probably at least a megohm. This is because the VOM puts a small voltage on the leads and the cap is initially discharged and begins to charge.....if it is good. When the resistance stabilizes, reverse the leads and you should see the resistance bang to less than zero and repeat the ramp up to a high value as it charges in the reverse direction. If it is open you will just see a high resistance always. Shorted, zero.
The other common failure mechanism in a capacitor start motor is cruddy starter centrifugal switch contacts. If there is one it is usually on the back end of the motor and usually easily accessable. If you pull covers you should expose all this stuff including another cap if it's there.
Hope this helps.
Al