Technically the backup Garret is referring to is from December last year, and is actually a backup made of your account on the old server before they were taken offline, thus being the state your account was in just as it was migrated over to DirectAdmin.
The backup is meant to restore the account in case of failure (includes emails, databases, files, settings etc), extracting specific files/folders is not as easy a task. Easiest way to use that backup is to wipe the account and restore from the backup which would reset everything to the way they were when your account was migrated in December/January.
That being said, your database should not have been removed when you removed the domain, only files should be affected, so you can probably get it working, minus uploaded contents, by simply grabbing the install package from wordpress.org, and instead of going through the installation process simply edit the wp-config.php to match your database information.