Hi @kirkyboo ,
Found two copies of the wiring diagram for a Kenmore 110 which are both virtually the same.
The wiring may NOT be the same as your model’s wiring, but you have to start somewhere.
(click on image to enlarge for better viewing)
Looking at the wiring for these models, the drum lamp should come on when the door is open, independent of the start switch.
Since you have replaced the door switch you must have an open circuit between the neutral input from the power outlet and the door switch.
The fact that it will start when you connect a lead to the frame indicates that there is a good connection between the neutral wire input and the frame.
You need to trace the wire back from the door switch to the neutral wire input connector block and check the connections etc.
Again the wiring may not be the same for your particular Kenmore 110 model