I do not have a Kindle device.
I have the Kindle App for my iPhone, iPad, and iMac.
The devices are listed under "My Account" > "Digital Content: Manage your Kindle" on the Amazon.Com web page.
When I buy books, it asks me if I want it delivered to my iPhone, iPad, or iMac. From any of my devices I can access any book I own by picking "Archived Items" and grabbing the book over the air.
You may want to try registering your iPad Kindle app to your account. That option is under the settings screen. (click the (i) icon on the bottom right of the Kindle App screen.)
If you want to turn on the slick page-turn animations, etc. in the Kindle App for iPad, click the (i) icon at the bottom right of the screen, Click "settings", and turn "basic reading mode" off.
The Kindle App on iPad is really well done. (Better than the Mac version IMHO.)
Oh, and while you'e in the Kindle store, by one of my wife's books!

Her name is "Wen Spencer" and she writes Sci-Fi and Fantasy books.