You can easily set up another $wpdb object to access your other database.
$mydb = new wpdb(DB_USER, DB_PASSWORD, DB_NAME, DB_HOST);
var_dump($mydb);
That, of course, uses the default connection credentials but if you defined different constants and used those, is should work just fine.
If this other database is not a WordPress database, you won’t have some of the functionality offered by the wpdb class but the basics should work if you write the SQL.