Lift out the upholstery staples holding the fabric to the chair's seat frame, on the chairs underside, to access the springs. Use a screwdriver to pry off the staples. Note on a piece of paper where the staples had been by sketching out the chair and marking where to replace upholstery staples later on.
Turn the chair right side up and pull back the fabric. Pry away any foam or batting, making sure not to destroy it.
Draw the configuration of springs and strings so that when you remove the broken spring, you know how to properly replace it. Snip the broken spring loose from its twine or string and set it aside. Shop for and purchase a similar replacement spring.
Install the new spring, using the upholstery staple gun to staple a new piece of twine or string to the chair base, and tie the new spring firmly in place.
Replace the batting and furniture foam as it was, and re-cover with the fabric. Using your notes and the drawing you made if necessary, staple the fabric back in place.