How do you fix the pull string on a Briggs and stratton push mower i keep having trouble with the coil?

Been there and done that. Please consider using eye safety glasses.

If you are up to a challenge, first purchase a length of nylon rope about the same diameter of the present pull rope about six feet in length. Next, remove the top of the engine cover by removing a small number of screw/bolts found along the sides of the cover. On the underside of this cover you will find the housing and recoil spring. Examine the operation of this spring by pulling carefully on the pull rope just a bit. If the spring seems to be still intact and not broken will will not need to replace it. DO NOT ATTEMPT TO REMOVE THE COIL! but first find some flexible small wire or use string to tie this coil to itself. (If you fail to carefully tie the coil together in 3-5 points around the coil you will be unable to replace the coil within the housing.)

After you have securely tied the coil tightly together you can then remove it from the cover housing and cut away the old pull cord. Now install the new pull rope with good knots. Don't worry about the length of the rope just yet, but wrap as much as the housing will allow. Be certain to wrap the rope in the same direction as the old rope was wrapped. Next, replace the coil spring and pull rope back within the housing and, after ensuring it is securely within the housing, then cut the small wires or string that has secured the coil. Now, go get a beer and pat yourself on the back. Money saved and the repair should last for several years more.

If the tasks set forth above seem too challenging, you will avoid a great deal of frustration by visiting a repair shop. The tension within the recoil spring is larger than you can imagine and it is a danger to you if it is not handled with care. If it comes free of the ties mentioned above then go get two beers and a band aid.