Help!-sinking 2nd Floor extension w/ no foundation/base floor

[My Nameish]

Hi, looking for a contractor (perhaps a foundation expert) who may be able to advise on the following: Looking at a duplex that has a little alcove extension (picture a breakfast nook kinda thing) on the first and 2nd floors--just a couple feet jut-out.  Underneath the extension is an outdoor "alley" walkway from the front yard to the backyard--i.e., there is nothing solid underneath this extension.  No basement, no pillars/supports, anything--and, because of that, the whole extension is sagging.  It has already caused a big bump in the floor on the first level, and a crack in the attic that has created holes and a leak into the 2nd floor when it rains, etc.  It appears to be rotting on the underside of the extension.

Can anyone recommend how to go about fixing this--or know someone who could advise?  This architectural design is common in Chicago, so I imagine someone out there knows something about this.  We are thinking it might be cheaper to remove the whole thing than try to figure out how to create support for it.