It sounds as if the battery isn't charging. Was that the problem with the old one? If so, verify that the LED on the AC adapter is ON before you plug in the notebook end. If it is not, replace the adapter.

If it is on, and the battery won't charge, it could be a faulty battery - but it's more likely a power jack or mainboard fault.

Not necessarily - it is often possible to repair the jack. You'll need a shop that specializes in board-level (solder) repairs for an evaluation and repair.

No, there isn't one. And bear in mind that a botched soldering job that would have cost $150 can quickly cost $400+ for a board replacement.