Sounds as though the "one-click-install" at Go Daddy (GD) put your WordPress into a folder named wordpress, hence the path. As we all know, when one extracts the WP package, it is, in fact, in a folder called ... wordpress :)
If the domain that you mention is the primary domain for your GD hosting account (the one the account was set up with), then as far as I know ( I have set up way too many at GD! LOL! ) that site needs to reside in the root of the account. What all this means is that you will have to move (copy, just to be safe) the contents of the /wordpress folder "up one level" to the root of your account.
1) Log into your hosting control panel and then the FTP File Manager
2) in the left pane, select the wordpress folder; you will then see in the right pane, the contents of that folder
3) click the check mark at the top left of the right pane (next to Filename) to "select all" (note, make sure that the Page Size (upper right) is set to 50 and that there is only 1 page available to be displayed)
4) with all the items selected, click the Copy icon and from the left pane, click the html folder when prompted as to where you want to copy them
5) if you get the notice xxx file already exists, click the Yes to all for overwriting (replace)
It will take a bit to copy them over, but you will get a Copy complete notice. Once that is done, then click the Databases tab and select MySQL.
1)once at the database page, click on the Manage with phpMyAdmin link next to your database
2) log in with the user name and password that was sent to your when you did the one-click install
3) once there, in the left pane, click on your database name
4) once expanded, in the left pane, click the wp_options table
5) in the right pane, click the Browse tab
6) in the right pane, click the pencil icon in the siteurl row; at the moment, your should read what you posted plus/wordpress; you need to delete the /wordpress from this entry then click the "Go" button (lower right)
7) click the Exit icon (upper left, left pane) to exit the phpMyAdmin
8) exit from the GD hosting control panel if you like
No need to edit the wp-config.php as the database connection info hasn't changed.
Your site should now be accessible by your_url and your_url/wp-admin