Yeah, that does look right. So it wasn't what I thought it might be, which would actually be an easy fix.
I have one other idea and if this doesn't work, reinstalling is probably the easiest way. Boot into the livecd and do.
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nano (where the livecd mounts the hard disk that you have mint on) /etc/sudoers
In this I'm assuming the livecd has nano. This is a tiny bit of a hack because normally you can only edit /etc/sudoers using visudoers, which wouldn't work because that command doesn't allow you to enter a path. But I noticed that Mint actually uses nano for this, and not vi. Anyway, the point is make sure the last lines of /etc/sudoers look exactly like this:
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# User privilege specification
root ALL=(ALL) ALL
# Members of the admin group may gain root privileges
%admin ALL=(ALL) ALL
If not, change them to match. Saving in nano is easy, just Ctrl 0 for write out. If this doesn't work, my guess would be that you somehow managed to take yourself out of the admin group.