The problem seems to be that the wiring are burnt out as seen in the picture.
I don't see any problems at all in that picture. The wiring is not burned out. Could you please indicate where you see any problem with the wiring?
What you've taken a picture of is a ceramic insulator. It's simply a porcelain piece that's meant to insulate the connection of the wires to the terminals of the surface element. It is highly unlikely that the problem is inside that insulator, but if you want to , you can check. Simply turn OFF that surface element, pry the two metal clips off both sides of the insulator and it will split into two halves. Take the two halves apart and check to ensure that the wires are tightly connected to the screw terminals of the surface element. If not tighten them. If so, put the two halves of the insulator together over the connection and put the clips back on the insulator.
While you have the insulator open, check for continuity across the two screw terminals of the surface element with an ohm meter to confirm that the surface element is good.
It's been my experience that when a surface element doesn't work, the usual cause is a wire burnt off at the switch, a bad switch, or a bad surface element. What you need to do is find the TWO 40 or 50 amp fuses to your stove in your fuse box and pull out the fuse holder they're mounted in. Or, if you have circuit breakers, trip off BOTH 40 or 50 amp circuit breakers to the stove.
Then move the stove away from the wall, turn it around, take the sheet metal panel off the back of the stove and look for a burned out wire going to the switch that controls that surface element. If you don't see anything, then you need to start testing to find out what the problem is. Unless you're familiar with using a volt meter, then fixing this stove would probably be best left to an appliance repairman.