Please help URGENT Why won't my laptop turn on?

My laptop will not turn on. Ok, about 8-10 months ago, my laptops power jack socket stopped functioning properly (slightly melted) so I opened the laptop and tried resoldering what I presumed were cold joints. To my disappointment, this did not work and my laptop has been idle for the last 8-10months (don't remember exactly when)

Recently, I bought an external battery charger in hope of reviving my laptop, however after charging the battery for a couple of hours (and the charger indicating it was at full charger) when I put it in the laptop, the laptop did not turn on. However, when applying the battery, one of the laptops LEDs would illuminate (signalling the battery is on/in).

Could the battery be 'dead' from a prolonged discharged period?

So my problem is, why won't my laptop turn on?
How can I fix this?

Thanks for any contributions

Brad

Ps the laptop is a samsung s3520 notebook