To update AIR on your KF you have to get root privileges. Also keep in mind that android build on KF doesn't have any copy command (it cut off). So the best way I found is to flash your KF with modified (pre-rooted) stock version and then install new air.
- Get pre-rooted stock version (I took it here). IMPORTANT: it installs via TWRP, google how to install TWRP on KF.
- Put downloaded .zip and air_runtime.apk (latest AIR version) in the root of KF.
- Reboot in Recovery mode (TWRP should load)
- Flash this version.
- On your PC open cmd and run "adb shell" (make sure you see your device in list when run "adb devices" otherwise check drivers).
- Run "su" (if you downloaded secure version).
- In shell go to sdcard ("ls" to get list of files/folders and "cd folder_name" to get into) and run "install air_runtime.apk /system/app/air_runtime.apk" (I think you can just run "install /sdcard/folder_with_air/air_runtime.apk /system/app/air_runtime.apk").
(7a. If it tells you that can't install because of file already exist, run the following two commands: "mount -o remount rw /system" (mount 'system' with read/write rights) and "mv /system/app/air_runtime.apk /system/app/air_runtime.bak" (rename air_runtime.apk into air_runtime.bak). Then repeat step 7.)
- In KF just run (install) air_runtime.apk (use any file explorer, e.g. download ES File Explorer from Amazon).
- Check AIR version in Applications.
That's all. Looks a bit complex, but in real it takes about 4-5 mins for me to update AIR (BUT I have TWRP already installed).
Hope it helps.
UPD. After your Kindle updates itself (version 6.3.1 currently the latest) you'll lose you SU privileges. AIR also will be rolled back to 2.7. You can prevent KF auto-updates (search on xda how to do it) or flash actual pre-rooted version (it gives you several months without problems).