/var contains some important folders which hold the state for programs (in /var/lib). Your program state (dpkg) has gone too, so you will end up with a reinstallation of programs anyway.
Since /var/log has gone too, you cannot analyse /var/log/apt/history.log either so you have to rely on your memory to know which programs you've installed. After restoring the dpkg and packages database (probably by sudo install -D -m755 /var/lib/dpkg && sudo apt-get update), you can use dpkg -S /path/to/file to get the package for a given path.
That said, a reinstallation would be the easiest way. Make a backup of /home (and possibly /etc if you changed system-wide settings or users/groups) and do a reinstall. If your /home was on a separate partition, it's even easier: mark that partition as /home in the installer, but do not tick "format".
