If you want to "officially" publish it in the Amazon App Store, there's probably no way around contacting Amazon and applying for "membership" for the
KDK. I applied there twice and never heard back from them, but if you are a company you may have more luck, especially if you have a concrete idea for the app you want to publish - which you do.
Depending on the complexity of the App, you may be able to do it "in-house" and re-use a lot of code from the Android version - apps for both Android and Kindle are essentially written in Java.