According to :
Thank you both for the prompt replies. In my usual fashion, I
neglected one fact:

The pull rope appears to be fine...just broken off at the recoil end.
There is still what appears to be enough rope to fix the problem, but
I may also be 100% wrong.

My plight is that I do not know where/how to physically attach the
rope to the recoil assembly. I realize that I am asking a lot, but I
really do not want to inflict any damage to the mower...although that
would mean that I would not be able to mow for a while...8-)

The rope wraps around a pulley. The pulley will have some sort of
attachment point where you'll probably need to tie the rope. The
direction you wind the cord will be obvious from the geometry
of the exit hole and the direction the recoil spring winds. Be _careful_
with the spring. Try to avoid releasing it. Recoiling it is a PITA.

As a FYI, this has a description of how you look for Tecumseh parts -
you need the "Spec Num" (Specification number).

http://www.landscapepower.com/tecumseh_partslists.html
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.