Macbook Battery Not Detected but stays on when unplugged

After replacing my MacBook 13" Late 2006 inverter cable, I could not turn it on when unplugged, even though the battery indicator indicated 3 out of 5 LEDs. I plugged it to the AC power and then it went on. When it booted, the Magsafe connector was orange and the battery icon was like this

0% Left (Not Charging)

Power Source Power Adapter

Time Left Calculating

After about a minute it went green, an X appeared on the battery icon and this was the information shown:

No Batteries Available

Power Source Power Adapter

The thing is, if I unplug the Magsafe connector, the Macbook would still stay on and would read this

No Batteries Available

Power Source Battery

Whenever I plug in the Magsafe, the LEDs don't flash slowly as they used to before. It seems like the battery only gives energy but doesn't take none from AC source. I tried to drain the battery but it would go off after around 20 minutes, with battery still indicating 3/5 LEDs, I tried to reset and update SMC, I did everything I could find on Google and Apple Discussions. No luck.

All I want to know if it's a computer, battery or software issue. Anyone got any ideas?

Thanks

Adam