My kindle will not open on my iPhone

Recently, Kindle app had many updates because of that problem.

Sometimes deleting the app, restarting the iPhone, then redownloading/reinstalling that app from App Store, solves the problem.

But if the issue persists, I suggest that you contact Amazon Kindle developer/support team and ask them. Use these instructions:--> How to contact an app developer - Apple Support

Regards

Giulio