How can I fix this hose winder?
It appears as though the portion that pulls out has sheared off from the inside. If you inherited this piece, I'm assuming you don't want to put much in it.

Clean it VERY well with some alcohol or similar (not acetone) and a brush. I'd try some J.B. Plastic Weld, let it setup and cure overnight and give it a shot. You've got a 50/50 chance it will either fail incredibly or someone will inherit it from you.

I've never had much luck with repairing things like this, but the J.B. Plastic Weld (and similar "plastic weld" products has fixed a veritable stable of Fisher Price toys, Big Wheels and similar.

Good luck!