Well that sounds like changes were made to the core files or possibly the database, as resetting the .htaccess like above should have solved that if it was just something re-written in there.
I’ve found a post which shows how to manually reset them, but it involves going into the mysql database http://tech.shawnlelle.com/resetting-wordpress-permalinks/
I would think before doing that you’d be better off re-installing a fresh copy of WordPress via the dashboard, disable all third party plugins as well. Then see if it gives you permission to edit the permalinks via the admin again.
I’ve never experienced the permalinks panel vanishing from the admin, I’ll ask the dev team if they’ve ever seen this and what they can advise further.