I dont mean to repeat, but I just got my GF a kindle fire, it came in about 3 hours ago and I got to hacking immediately.
This manual root method is outdated, the TWP link is to 2.1 and does NOT auto-update FFF. If you use this method, and rename to TWP.img correctly, you will get TWP but upon reboot as stated you'll enter a loop at the "Kindle Fire" screen. If you get a "unknown device" you need to have the drivers pre-installed, have th kindle already connected to the PC and then reboot the Kindle. If you boot the Kindle while its already connected to the PC it SHOULD recognize in the drivers as a kindle, how ever if you disconnect the USB and then reconnect it it will show "device uknown." My Fix was:
-power down kindle, attach to USB (it will boot when you plug in, power down and reboot)
-Your PC will attempt to install drivers (and should recognize it as a kindle) but fail
-open device manager, right click>properties>drivers>uninstall (and click the check box to "delete driver etc)
-download KFU .9.5 and run >install_drivers
-power up, device manager, right-click>properties>drivers>update drivers>choose driver>manually choose a driver from a list/have disk and navigate to the KFU.9.5 drivers>kindle folder and choose android_winusb
-install the driver and it should finish reading as android adb etc.
-KFU .9.5 bootmenu> option 3 (recovery)
-the device should respond almost immediately and factory reset to stock
-when booted back to the OS, run KFU FFF, KFU root and SU, KFU (custom recovery of your choice)
It took me god damn 3 hours to get it right, I thought I bricked it as I couldnt get a CMD ADB bridge to manually find the kindle, KFU pushed through the fastboot commands even though an active ADB wasnt running. good luck!