Why doesn't the Grid Fill Tool work on this mesh? I would like to create faces within the selected edges that match the geometry outside. 
In case that is not possible at all using Grid Fill Is there another way to do that?

 


For me it doesn't look too different to the situation shown in the documentation:




.blend
As Gandalf pointed out the area can be Grid Filled by:


Separate V the selection 
Grid Fill Space,Grid Fill or Ctrl-F / Grid Fill
Remove Doubles Ctrl-V / Remove Doubles