As of now the latest Firmware that can be Jailbroken is
5.16.2.1.1
As new Jailbreaks come out hopefully they will be added to the
Kindle Firmware Wiki page.
There is a Hardware Jailbreak called
Popcorn that I believe works regardless of firmware, but it requires opening the kindle, connecting some pins, or otherwise doing some physical things that, if done wrong, could ruin your device. I haven't got around to adding that to the jailbreak Table, but plan to at some point when I have the time