Site running from accidental upload of core in subdirectory
I had a similar incident where someone had put a second set of core modules in a second folder called "modules" inside the main modules directory.  As a result Drupal was running off of the system.module located in modules/modules/system.  This made updating the site problematic.

The only way I fixed it was by dumping the database as a text file (.sql), editing the modules/modules file paths in the systems table, then re-importing, overwriting the database.  You may also be able to do this via MySQL commands or using an interface such as phpMyAdmin.  You should also look at my other answer about steps taken when editing the database. (You would almost think that cowboy edits of the database is all I do everyday)

Know that editing the database is not best practice, but sometimes it is the only fix.  Make sure that you should have a backup in case you mess things up, which in most likely happen on our first attempt.