[Q] KFSOWI root with actuall Kindle OS 4.5.5

Hey guys,

my girlfriend is owner of a Kindle Fire HD (kfsowi) i wish to root it, cause we wanna use our chromecast and google play.
I rooted some devices before and it worked like a charming for me. Sadly I didn't found anything about routing the actuall version 4.5.5.
I did a fastboot cable by myself and it works, the kindle goes into it.

Thanks a lot