It's possible that the element is bad, causing no heat. It's also possible about a half dozen other things are wrong, from a loose or burned wire, to a bad thermostat, etc.Not shutting off won't burn up the element on it's own.

There should be a coiled round spring that fits around the filter's short tube. Maybe it's on that spindle in the drum? Lost? It ain't rocket science, you can probably get some sort of spring at Home Depot

here's the final filter assembly