What type of capacitor to buy
This is a very common Motor-Run cap  worth about $10~$15 or more for higher voltage ratings.
Search 22.5uF metallized polypropylene radial capacitor 250Vac (min)   5%
e.g. M72A2822N22 AEROVOX  280Vac 5%
or another
https://www.mouser.com/ProductDetail/Panasonic/EZP-Q38226MTA/?qs=TiOZkKH1s2QJT3hK7a6J8Q%3D%3D
You can solder to wire leads and silicone or PU adhesive to secure.