Cyanoacrylate can't handle nonporous materials - as you're seeing, it dries so brittle that an incautious glance will snap it.

Try epoxy instead; it's slower to cure and smells funny, but produces a much stronger bond, and doesn't have the same problems bearing mechanical loads. I like J-B Weld and J-B Kwik, which are strong and resilient as hell (you can fix tractor engines with them!) and double as a low-temperature casting material for additive repairs, but for what you're dealing with here, the cheap Loctite stuff on the drugstore blister card should do you.