Amazon went to significant trouble to make it practically impossible to extract a kindle file out of the Kindle app for ios. (They even make it rather hard -- though not impossible to send files directly to the device app without using the cloud). You can use itunes to add a file into the IOS app (using the "File Sharing" function in itunes; connect the device, select the device --> Summary --> Apps --> (scroll down and select Kindle app and then choose Add file).
Maybe (hopefully) Calibre lets you add files to the ipad. But this is a one way street. Amazon wants to make it impossible for you to have direct access to the ios file system.
But you may appreciate my perspective.
The Kindle app for ios is a truly wretched app for rendering ebooks. As an ebook publisher and formatter, I spend a huge amount of my time trying to make the ebook look less-than-crappy on the Kindle app for ios. (by comparison, It's a breeze to format for Kindle on android or the Kindle devices). Most of my ebooks just look horrible on the kindle for ios. I make a lot of compromises.
The rule of thumb is to try to use whichever ebook reader is native to the device. Ibooks is a best-of-class ebook reading software and certainly the best ebook reading app on ios. If you're buying from a third party (Smashwords?) you should have gotten an epub version and just emailed it and opened it in ibooks. It would look beautiful!
From a practical point of view, I sometimes read purchased ebooks from Amazon on my ipad app. That is unavoidable. But for reading third party stuff on your kindle app, you should be using the Send-to-Kindle function to get into the cloud, then go to the website and choose the device you want to deliver it to.
Your question is seeking to break through the Kindle app's DRM. Besides being possibly illegal, it also will be technically hard to do. I realize that you probably obtained the ebook file legally. I am unclear what you mean by sharing function.
Perhaps you should log in to the Kindle Personal Documents service via computer, go to "Manage your content and devices" , for show, select Docs, select the file and click the Actions button and then choose Deliver from the popup. That's the primary way you share Kindle ebooks across devices.
Finally, a personal recommendation. Even though I have never bought an ebook from the Google Play store, their Play Books app is phenomenal and is the best share-across-devices-and-platforms solution. Works great on ios, Kindles, androids, Windows. They have an easy-to-use interface for uploading epub files to the cloud.