There will be a black cased rectangular capacitor in there. It will have either 2 or 3 leads, and will be rated from 1.2 to 2uF, if 3 leads there will be 2 sections, one as before and another of 0.5uF. Voltage rating is 440VAC. You can get one out of an old ceiling fan ( one scrapped for broken blades, not because it will not turn) or buy a new one from appliance repair suppliers. I pay under $1 for them, and they do fail often. To test the old one disconnect the leads and check capacitance. If lower than 10% from rated then replace, and lubricate the bushes as well, with a drop of engine oil on each bush.
I have a lovely ceiling fan going to recycling soon, shorted windings to frame. Only noted when it stopped turning, and was touched ( no earth to light rose).