KindleBreak is the easiest jailbreak in years and works for FW versions >= 5.10.3 and <= 5.13.3
https://www.mobileread.com/forums/sh...d.php?t=338268
WatchThis works for 5.14.2 and lower, but how low it can go depends on which kindle model. Some people have had trouble following the steps for WatchThis.
https://www.mobileread.com/forums/sh...d.php?t=346037
There is a PW3 jailbreak for FW 5.9.6.1 that requires downloading and installing a large "factory" firmware.
https://www.mobileread.com/forums/sh...d.php?t=313086