You will need to attempt to flatten the top of what is left of the pull shaft, either via an razor blade/sand paper. This may be difficult depending on how far down the inner shaft is. I used a pair of needle nose pliers originally to pull the shaft up and held it in place with a sharp screw while I sanded the top down.
Then use a 1/16” drill bit and drill a hole ½” into the rod - be sure to drill straight down into the rod - perpendicular to the horizon (assuming the unit is mounted). Also note: the hole/drill bit/screw in the example below is bigger than 1/16”, but the screws supplied with your knob are smaller so please use 1/16”. Your depth of your hole may vary depending on how much knob has been chewed off. Start off with a shallow depth, and extend as needed.
Install you optional body cover protector over the unit before this step. Pull the pull cover over top of the damaged housing, then pull up the shaft using your screw and hold the shaft in the extended position using a screwdriver or drill bit - much like you had to do to sand/level the top of the shaft. This is to keep the pull rod extended so that you can screw the knob on - you may need an extra set of hands to do this.
While keeping the shaft pulled up with pressure against it so it will not retract, unscrew your screw and put it in the replacement knob and screw it into the hole you drilled.