Find someone who fixes lawn mowers, and they may sell you a used starter assembly. The assembly usually is held onto the starter housing by two little 1/4 inch hex-head screws. If your mower has lots of fakey plastic stuff on top of it, you may have to remove some to get to the starter screws. I sell used starter things for $10, plus installation. If you have any mechanical smarts, you can usually replace the part yourself.

I decided to try WD-40 and it worked. Was reluctant to put it on the starter parts because it can attract dirt and grime - but it must have slicked it up enough for the recoil to work.

The center sleeve was gunked up with dirt and maybe oil.

Actually WD-40 wasn't a bad choice because it's not very oily and dries up pretty fast. Ideally, the parts would be blasted clean with brake cleaner spray and possibly some graphite used on the recoil spring.

Graphite dry lube would be the best as suggested by Sax , since it will not leave residue which will attract dirt / dust . Actually if you go to a local jobber Molybdenum Disulfide Spray will do a very good job within this requirement . I used quite often back in my younger days when Dirt Bike Racing for Chain Lubrication . It goes on wet and via capillary action creeps into any pore or crevice and then drys completely leaving a well lubricated component without any tacky film .

You are apparently able to remove and rewind the rope. Just pre-tension the pulley by a few turns then re-wind the rope. Much as you would do with a vacuum cleaner power cord.