how do i restore purchases after updating ipad?

As described, the originally installed Kindle App is unlikely to be compatible with the iOS 5.1.1 system software update that your have installed. As you are unable to update the Kindle App on iPad1, you’ll not be able to recover access to the previously installed Kindle books.

Note also that you cannot regress to iOS 4.3.

Purchases for what - Apps, Apple Services, media, something else? Updating an iPad rarely requires any purchases to be restored - unless you have completely wiped/reset the device.

For Apps, media and services purchased from Apple, you simply need to be signed-in to iCloud and App Store using the same AppleID from which the purchase was made - as all purchases are forever tied the associated AppleID account.

You have an original first generation iPad. This model is obsolete - all support having been withdrawn around nine years ago.

You’ll not be able to download or reinstall Apps, such as Kindle, from the App Store - and similarly may have difficulties with other content and services.

You need to consider replacement of the iPad1 with a much more recent model of iPad. Even the entry-level iPad9, just released, will run the majority of available Apps - and will have reasonable expectation of full update support for four or more years. The previous (and now slightly cheaper) comparable model, the iPad8 is also still widely available.

Where compatible Apps (e.g. Kindle) are available - purchased books, media and subscriptions should be accessible by the replacement iPad. Just ensure that the mew iPad is set-up from new with the same AppleID - and for external content, such as Kindle books, that you sign-in to the original account that was used to purchase the books from the bookseller (i.e, Amazon).

Any updates to an Operating System can adversely impact existing software applications; this is not unique attribute of Apple or iPad, but applies to almost every computer system. The Kindle App was not developed by Apple - and therefore the App has to be maintained by the App Developer - which is in fact the case.

You issue is that of age - and support. Update support for iPad1 ended in 2012. The Kindle App had been updated to continue working with the iPad - and had the iPad and App been updated whilst both were “in support”, you wouldn’t now have the issue that you have.

In reality - and whilst I recognise your anguish - the problem is not of Apple’s making. Your problem is purely due to timing and limits of available App support.