If you are willing to update to 5.10.3 you can use kindlebreak
https://www.mobileread.com/forums/sh...d.php?t=338268 which is one of the easiest kindle jailbreaks ever. Afterwards, you can downgrade if you like.
The factory methods were better than nothing, but they involve very large firmware files and are specific to the kindle model. In general (very few exceptions) you can not downgrade kindle firmware, so you can not use a factory method on a higher firmware and I don't remember any firmware methods for 5.10.x and higher.