The braided part is the ignition ground. It should always be in contact with the engine mass. If not, the ignition may have failed. It can fried by high tension pulses if a earth return path is not present.
Restore the ground path in the cap, and check function. You may be lucky.
From your reports, it seems that this engine has seen quite some extra wear and tear in connecting the muffler and the ignition cable lay-out. Check the braided cable for further damage. Support the cable in such a way, that it cannot get damaged in any way.
fasten the muffler without gasket, and use some silicone RTV for tightness. If you do use a gasket, check the tightness when the engine is cold. The gasket tends to compress, and if loose, will damage the threads in the cylinder. For that reason, it is good to keep the cylinder studs well mounted using locktite.