First of all, make sure you
actually have the cage with you (should be in the key items section of your inventory).
To do so, you should have uncaged ryn at this point (complete Demon of Ice with ryn caged, or talked to Lumina after Shadow over the Sun)
Second but not least important,
make her submissive enough for her to let you put it in her (Otherwise she'll attack you. it won't damage you, since it's more of a flavor text, but the cage will still be on your inventory)