@henriko, you mentioned that two of the three speeds work. Is the non-working speed the slowest one?
I'm thinking that capacitor has simply gotten so old that it's not working as well as it should. It's going to take more current to start up the motor at the slower speed, whereas at the higher speeds there's still enough current to turn the motor. The weakened capacitor can't store enough current to drive the motor in its high current demand, but still has enough to run them at lower currents.
I had a similar situation with my air conditioner. The fan wasn't running, but if I turned it on and manually started the blades it would spin up and run fine. Replacing the startup capacitor fixed the problem. If there's a way to do it without risking your fingers, can you try spinning the beater(s) manually and see if they start running on their own?
That is almost certainly a paper capacitor, commonly used in antique radios of that era. In those days, the paper worked fine as long as it wasn't exposed to moisture, which is why they were encased in wax.
A paper capacitor may look gooey or melted, yet test OK. Conversely, it may look perfect, but test bad on a capacitor checker. The leakage is internal, caused by water vapor that invisibly penetrates the wax or plastic coating over time, no matter what the exterior looks like. Melted wax is often the result of decades of normal operating temperatures inside the radio.
When repairing antique radios, the first thing that is almost always done is to replace all the paper capacitors, and I see no reason why your vintage mixer should be any different. That last digit on the label is almost certainly "/ 4 A", which really doesn't matter in finding a replacement. Basically you're looking for a 250 V / 0.1 uF capacitor of most any kind except electrolytic. Electrolytic caps are polarized; the have a plus and minus side, where the paper ones don't, so you don't want an electrolytic. Here's an example of a kind that the antique radio buffs prefer.
uxcell 10 PCS 250V 0.1uF 2 Terminals Lead Rectangle Polyester Film Capacitors - - Amazon.com
Hope this helps; let us know how it goes!