From internet searching, it seems like your bookcase has had its day if you want to get rid of the silverfish (though from what I've seen so far, they don't appear to be actually dangerous), as once an infestation has taken hold, it seems that realistically it needs harsher chemical treatments to irradicate.
Cinnamon, cloves, lavender, citrus oil, basil & cedar wood (which is usually on unsafe wood list!) appear to repel them. If you google 'silverfish aviary' you will find products such as this: Control Natural Aviary and Cage Bug Spray
Clear out piles of old newspapers & cardboard which they hide in and eat.
Transfer any foods in cardboard packets into plastic containers.
They thrive in dark damp areas - there's not much you can do about humid air in an aviary but consider if you create other sources of damp e.g. Put bathing water containers on a plastic tray so spilt water doesn't soak into surfaces.
Sweep & vacuum regularly to help remove eggs.
Diatomaceous earth is mentioned a lot but with advice to wear a face mask when applying, so that's also likely to be unsuitable for the birds.
Best of luck!