Insufficient Storage Available when installing Google Play Services I have rooted my Kindle Fire 7 (4th Generation) and installed Gapps using the Jmz Fire HD7 Tool. Everything went well, but, when I was trying to start Gmail, it asked me to update my Google Play Service. When the installation was nearly finished, a message "Insufficient Storage Available" popped up.

What can I do? I have tried to clear all data and force stop Google Play Service, and it still can't be updated. Any help would be appreciated.

EDIT

Pasting the result of df here... looks no good.

Filesystem               Size     Used     Free   Blksize
/dev                   435.9M   288.0K   435.6M   4096
/sys/fs/cgroup         435.9M    12.0K   435.8M   4096
/mnt/asec              435.9M     0.0K   435.9M   4096
/mnt/obb               435.9M     0.0K   435.9M   4096
/storage/usbotg        435.9M     0.0K   435.9M   4096
/storage/emulated      435.9M     0.0K   435.9M   4096
/mnt/secure            435.9M     0.0K   435.9M   4096
/mnt/asec              435.9M     0.0K   435.9M   4096
/mnt/obb               435.9M     0.0K   435.9M   4096
/system                  1.1G     1.1G    23.9M   4096
/cache                 852.4M   483.7M   368.7M   4096
/data                   12.3G     1.3G    11.0G   4096
/mnt/sqfs              124.6M   124.6M     0.0K   32768
/persistbackup          10.8M   184.0K    10.6M   4096
/mnt/shell/emulated     12.3G     1.3G    11.0G   4096
First, turn off power on and off your phone and see if the problem still exist . If so,
Go to your storage usage located in your settings and try deleting one unimportant app 
If steps #1 or #2 didn't work or go through, just simply unmount the SD card the correct way. Don't just pull it out. If you have been doing this, it is probably why it's doing that now.
If you have done these steps, it should work. If not, go to a little side store that fix phones or the mall. You can also go to your service provider and see if they can fix the problem. Otherwise, go to amazon.com and look up SD card installment kit and you should be able to pick which one you want from them.