Its a very old thread, but I don't see that the solution is here or the this thread is marked as solved. I have also stuck into the same problem I solved it like this
$path = public_path('../storage/YOUR_FOLDER_NAME/YOUR_FILE_NAME');
if (!File::exists($path))
{
File::delete(public_path('storage/YOUR_FOLDER_NAME/YOUR_FILE_NAME'));
}
The key is that you need to remove '..' from the delete method. Keep in mind that this goes true if you are using Storage as well, whether you are using Storage of File don't for get to use them like
use App\Http\Controllers\Controller;
use Illuminate\Http\Request;
use File; // For File
use Storage; // For Storage
Hope that it will help someone.