My "shut-off" cable and bar came off... how do I reassemble this apparatus?

Lots of missing information.

Look up the model of your lawn mower at www3.sears.com and find out what pieces are missing. The screws and nuts are likely missing. Use their diagrams to re-assemble.

Place the tang on the cable through the hole in the bail handle, then put one end into the handlebars, then press the other side just enough to allow you to put the other side of the bail handle into the other hole.