I would agree / is very small. It seems that apt is running out of space in /var/cache/apt/archives. I think the long term solution would be to make / bigger, but this may not be easily accomplished. You may be able to "borrow" some space from /home by doing a bind mount. I've tested this locally to install a package, but I can't make any promises.
1) Make a directory in /home
# mkdir /home/archives
2) Do a bind mount of /var/cache/apt/archives .
# mount --bind /home/archives /var/cache/apt/archives/
3) Re-run your command to get the new kernel.
# apt-get -y install linux-image-2.6.32-5-amd64
4) Unmount the bind mount, so things are back to normal.
# umount /home/archives
Good luck, and the usual disclaimer