In case this issue is still unresolved, your wife needs to link the kindle app she downloads to her Amazon account. The easiest way to do that is to download the kindle app from within her Amazon account on the full Amazon.co.uk website.
The procedure is straightforward but has several steps - see below:
1) Log into her own Amazon account on her iPad using the full Amazon.co.uk website.
2) Hover over the "Hello,.... your account" on the black banner at the top of the page and a drop -down list appears - scroll down to "manage your content and devices" and click on it.
3) Just below the black banner at the top of the screen should be a list starting "Kindle Store" "Buy a Kindle", etc., the last but one in that row is "Free Reading Apps" - click on that.
4) She should now see a "GET THE all-new Kindle App" logo which has black buttons to download on the Apple App Store, Google Play or Download for PC & Mac. Or she can enter her email or mobile number in the box on the screen to deliver the kindle app to her device...