I can't tell if that is a mica insulator on the bottom or galvanized metal. It looks like you could have spilled something on it.
With soldering iron tips and hot air rework stuff, I usually just cycle them to their highest setting a few times to get rid of any smell. I'll do it outside if it is particularly bad. I have a solder pot that is really bad and scary as hell when there are a few ounces of liquid tin/lead melted inside.
If you have a multimeter you can check the connection to be sure there is not a short between the chassis and element, but that is quite unlikely. If it has no ground, touching the thing would be... invigorating... if there was a short circuit. If there is an earth connection, it would trip your circuit breaker.
Mica is kinda a pain to cut and it makes a mess. It is a cheap high temp insulator. You should be able to buy a sheet or maybe even a disk that would require minimal rework to replace the one you have.
Personally, I'm cheap and would just head to a thrift shop. Most consumer junk with a heating element will have a mica insulator sheet or few. Something like a $1 toaster should have something you can physically see without taking one apart in the store. Most cheap toasters have mica sheets with some ni-chrome wire wrapped around the sheet(s). It is probably fine as is, and the smell will likely dissipate in time.