Further to what Duncan says.
A time delay relay in the start winding feed is an easy way to get it going if the centrifugal switch is dead. A couple of seconds before the relay turns off power to the start winding should do it.
Time delay relays are bog standard and not silly expensive components these days. DIN Rail mounting ones from "proper" UK suppliers start around £20 + VAT, imports via E-Bay from about half that. If you aren't an electrical / electronics guy with a big box of stuff its hard to beat the price by buying bits and making something.
However when I did similar for a 3 hp motor with no internal centrifugal switch I just used a monster pushbutton switch alongside the ordinary DOL start button to bring the start winding in with a two finger "count 1 & 2 & 3, she's off" two finger push. Simples if not strictly kosher. Nowadays I'd get a time delay relay and do it properly but 30 + years back such things weren't affordable.
Clive