the 2nd pic shows what looks like the end of the cable.
the 3rd show the roller and hole where it comes up.
to the left it show a groove cut in the top, this counties to the lid. at the end of the groove, should be a screw/ pin the cable hooks to.
what happens is the lid pulls the machine up as you open the cabinet .
I would use a good nylon or ?? a string you trust. install that in place of the cable .
Do this with no machine in the cabinet. open and close. adjust the string ( or buy small cable) length until it works just perfect.
then measure the string for length. start calling place that may like hobbies stores, clubs that deal with hang gliding , home built aircraft, maybe a bicycle shop ( great place for cable stuff ) that caters to custom bikes. even an electrical supplier / repair guy.
your looking for someone that has some type of hi pressure crimper