I have an old iPad mini with the Kindle app that my brother downloaded e-books on. Now I have a new iPad mini 4 and I can't seem to transfer the books to it.
I haven't bought these books on Amazon because they were downloaded from my brothers computer.
There's no way you'll be able to transfer the e-books from the old iPad mini to the new iPad mini 4. You'll at-least need access to the Amazon account that was used to sign into the Kindle app on the old iPad mini.
Kindle allows you to read personal documents and e-books which have been uploaded to the Amazon personal document storage service (without purchasing any e-book or paying anything). The content can be synced into the Kindle app installed on any supported device. But you'll need the login information of the Amazon account at a minimum.
Even if you manage to backup the old iPad mini and restore it onto the new iPad mini 4, the Kindle app will be installed alright, but you'll still need to log into the new iPad mini with the Amazon account credentials which was used to setup the iPad mini.
Once you manage to register the new iPad mini 4 by logging in, you'll have access to the entire library of e-books and personal documents linked to the Amazon account.
To make it clear, the Amazon e-books and personal document content is no way related/linked to your Apple ID and no corresponding data is store in iCloud/Apple's servers.