The KingRoot rooting utility is now capable of rooting the Amazon Fire TV 2 running software version 126.96.36.199. Software update 188.8.131.52 blocked the previous rooting method used by KingRoot, but now the app has been updated and is capable of rooting software version 184.108.40.206. Several people on XDA have confirmed that v4.9.6 and v5.0.1 of KingRoot was able to root their Fire TV 2 running 220.127.116.11.
It is likely that KingRoot will also be able to root software versions 18.104.22.168 and 22.214.171.124 on the Fire TV 2, but I have not seen confirmation of that. Several people have tried using KingRoot to root the Fire TV 1 and Fire TV Stick 1 running 126.96.36.199, but nobody has succeeded. So this means the Fire TV 1 can still only be rooted if it is on 5.0.5 or older, and the Fire TV Stick 1 can still only be rooted if it is on 188.8.131.52 or older. KingRoot also cannot root any device, including the Fire TV 2, if it is running software version 184.108.40.206 or 220.127.116.11.
I will update my rooting starter’s guide soon to include this new information, as well as the recently released TWRP and ROM for the Fire TV Stick 1. If you do root the Fire TV 2 using KingRoot, it is highly recommended that you immediately install TWRP, perform a factory reset, and then install the latest pre-rooted ROM.