The only file in your root WordPress folder that you should need to keep (given your screenshot) is the wp-config.php file, unless you've got non standard customisations to the site. You should be able to delete the wp-admin and wp-includes folders (these are the "core" folders David mentioned) and reinstall them from a fresh copy of WordPress.
Cleaning a site isn't easy. You'll very likely have infected files in your plugins, and as David mentioned it's quite a process to remove and reinstall them. On top of that, you'll need to:
* reset all admin user passwords in the database, and probably editor and author user passwords too (and maybe more depending on the site)
* change the database user password on the server and in the wp-config.php file
* change the "authentication unique keys and salts" in the wp-config.php file (there's a note in the file on how to do that)
To be completely thorough, the above should be done offline and without opening WordPress, to avoid reinfection part way through. There may also be other things in the WordPress database that are compromised and open the site up for reinfection. I've found that generally I can clean a site manually without being completely thorough, but it does take quite some time.
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