Okay, this is going to be fun. Your box is fairly screwed, so the worst problem is that you screw it harder, but I take no responsibility for you doing that following this advice. The problem is either that the dpkg package didn't install correctly, or the metadata is corrupt.
Let's deal with the first problem. Debian packages are just an ar file of two tar files. You can extract them by hand if you really need to. Try the following:
# cd /tmp
# ar x /var/cache/apt/archives/dpkg_1.13.26_i386.deb
# cd /
# tar -xzvf /tmp/data.tar.gz
This should extract the files from the package and install them on the system. We should probably tell dpkg that the package is installed. Find the /var/lib/dpkg/status file, and find a line that says:
Then edit the Status line to say:
Status: install ok installed
and change the version to say:
Hopefully now, dpkg should be working. If it is, I would reinstall dpkg, so dpkg knows what files the dpkg package has installed.
# dpkg -i /var/cache/apt/archives/dpkg_1.13.26_i386.deb
If it's a metadata file that's become corrupt, they're just plain text files, so you can edit them. I would suggest using dpkg in debug more or strace to see if you can find out which file in particular is broken.
# dpkg --debug=2001 -i /var/cache/apt/archives/dpkg_1.13.26_i386.deb
# strace -efile -f dpkg -i /var/cache/apt/archives/dpkg_1.13.26_i386.deb
(Check out dpkg --debug=help to find out what 2001 means)
If the status file is corrupt, there is a status-old file you could use as a backup.