If you can still ftp into your WP site and the files are still present, there’s a very good chance your database is still in tact.
To find your database, FTP into your WP files or open your backup copy. Within the files, use Notepad to open a file called wp-config.php
Once opened, look for some code that looks something like this (usually about 20 lines down from the top)
/** MySQL database username */
/** MySQL database password */
/** MySQL hostname */
What you need is the MySQL hostame - something like mysql.mysite.com
Copy it into a browser and press enter. When asked for a username and password use the ones your code (like above) and voila you’re now into your database.
Now all you need to do is press the Export button to proceed to the area to download your database.
When you first open wp-config.php it may look very confusing if you’ve not worked with code before. The best thing to do is just calm down and study the code for 5 minutes or so and everything will fall into place.
Hope that helps.