Not sure how/why your /etc/sudoers got set to 0777, that is not what you want--0440 is the correct permissions. Looks like with the incorrect settings it is not even allowing you to use the sudo command. You might want to try logging in with single user mode. You can do this by hitting ESC during boot to get the GRUB menu. Hit 'e' to edit GRUB, then choose the line that begins with kernel and hit 'e' again to edit that line. At the end of the line type in 'single' (without the quote marks). Hit enter, hit 'b' to boot. This will boot you into and terminal session as root without asking for a password. From here do the following
chmod 440 /etc/sudoers
You can also set roots password here by typing
And entering what you want for the root passwd.
Once you have reset the permission on /etc/sudoers and done what else you need to do from this session all you need to to is reboot to revert back to your normal login.
I would follow duryodhan advice though and not setup a root password, but use sudo instead. If you have several commands you need to run as root and do not want to have to type sudo before each one just do one of the following.
Hope this helps.