How to get books to iPhone kindle app?

I had a kindle but it broke (screen cracked) I'm trying to retrieve the books I purchased on my kindle on my iPhone. I went to manage my devices but I don't even see my kindle on there and it was registered. I had an android phone with the kindle app and everything worked fine so I do not see why the iPhone app would not work. Any ideas on what I can do to get back the books I paid for?

You go to archives and tap the books you want. They will take a minute or two to download but as long as you have wi-fi or a signal you should be able to retrieve you books. I don't think you can buy books off the iPhone though unless they added that to the app (I know they did for the iPad but not sure about the iPhone).

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