Scrape off the gray junk over the AC adapter port and follow the traces. You should be able to see the traces as to where positive and negative are supposed to go. You might only have to retrace the positive or the negative. Meaning you may not have to tap both.
Replace the dc jack, it should fix your problem. The dc jack has a switch that un hook the connection from the batteries when dc is plugged in. Over time, that switch goes bad.