I would not even bother reattaching the squirrel cage, when a replacement plastic cage is available and cheap. Mine came apart during restoration too, and my attempts to reattach it were unsuccessful. It did reattach, but then wobbled like an out-of-round wheel until it shook itself apart again. It's not worth it. Just buy a new squirrel cage.
By the way, most auto electric shops have the equipment to separate the squirrel cage fans from the motor shafts, and also sell replacement squirrel cages for 20-something bucks.
