Clearly there was a short within the logic board likely a shorted out MOSFET. At this point you have issues with both the logic board and your battery needs replacing as well. Its going to take some effort to fix.
Which model do you have? If you have a newer Core i7 (2760QM or 2860QM) with AMD Radeon HD 6770M GPU, I would spend the money to fix it as this is a treasure of a system! This is the last 17” model Apple made and is sought after by quite a few types of users.