That's exactly what git add -p does, and it's vastly more convenient than when you don't have a staging area.
git add -p doesn't help tease them apart, because its hunks are based on diffs, and the granularity usually does not coincide with actual changes (nor syntax or semantics of the languages). Which is really expecting a bit too much! Diff's implementation of LCS algorithm is incredibly fast and works amazingly well for changes in different places.
So what I do is edit the file to what I want, stage and commit, then undo and continue. This doesn't really use the staging area, and I'm hoping there's a better way to do it... if anyone can think of one?