Hi, I'm Pete. Welcome to JustAnswer! I'm reviewing your question now, and will post back with your reply ASAP.
You will need to add the iPhone as a new device on your Kindle account.
Open the Kindle app on your iPhone and sign in to your Amazon account.
Let me know if that works?
Can you install it from the app store?
Did my answer work?
Ok, is there anything else I can help you with today, or would you like this question cancelled?