The PW2 is one of the most jailbreakable kindles, but I have no experience with it. You should open a new thread with PW2 in the title and maybe a PW2 expert will respond. You should search for threads on PW2 jailbreak, which might have all the information that you need, or at least give you some background to make it easier to communicate with the experts.
If all you want to do is update the firmware, the thread below tells how to get any version for the PW2 in particular. Be sure not to go too high because of UI changes that a lot of people don't like and more aggressive defenses against jailbreak.
https://www.mobileread.com/forums/sh...d.php?t=351683
the page below lists all the kindle firmware versions for each model.
https://www.amazon.com/gp/help/custo...Id%3D200203720
jhowell,
please note that PW3 came out with 5.6.1.1