Ok , since the warning light dims out either you have bad battery/low voltage or bad earthing or damaged cables as you mention and finally faulty starter.
Also , since you mentioned about the melted cables , check for broken fuses if any.
First of all replace melted cable(s) and check for short-circuits and burned fuses. You have to be sure that there is no naked cable , especially a "live" one touching ground if you dont want fire later. When finish with that and try once then check again fuses. If there is another burned then the problem exists and need to check more. Also try with a battery booster or another battery or just charge this one just to be sure it is not voltage issue.
There is also a posibillity that the starter is not working properly. I had similar problem , it didnt not kick and warning light was lowering. We tried direct voltage from battery , still didnt kicked in or it kicked 1 out of 3 so i replaced it with another one and worked fine. Probably previous attempts caused overheating cables and melting them due damaged starter.
Sprint 1.7 '85