The spring loaded part that is attached to the cord is the starter assembly. If you are able to pull the rope and it has tension and then snaps back when you let it go then this component is likely working properly.
If the starter is apparently broken, I recommend that you have a service technician replace this part. Here is a link for the Sears Service website that has drop-off locations:
Sears Home Services
If the starter is working properly, I recommend that you try draining the gas tank and refilling the weedwacker with fresh gas and oil mix. Pull the spark plug out and make sure that you are getting compression in the cylinder. Check the spark plug and clean it if necessary. Make sure that the gap is set properly. Replace the spark plug and attach the wire securely. If the weedwacker still won't start, I recommend that you have it serviced (or replace it).