How can I triage a MacBook Pro that won't charge with the LED dimly lit green? I have an early-2011 MacBook Pro that will not charge. The green light on the MagSafe connector is barely green. 

Besides the fact that the system info tells me the battery is not charging, the battery is stuck at 39%. If I unplug the charger, the battery begins to get low and I soon as I plug the charge back in the battery stops dropping. 

Some sites say that the problem may be in the logic board and some say that the problem may be in the battery itself. My notebook is always plugged and I never really had a chance to let the battery drain completely. 

The system profile shows the battery condition as normal and that the ac adapter is not connected.

So, my question is: what should I try since my city does not have an Apple Store that I can take my notebook to? Is it possible that letting the battery die (forcing a cycle) could help in that case?  Is there any way of testing if the problem is in the battery (since it is more affordable to change the battery than a logic board)?

EDIT

What I have done so far:


I have already left it to charge for several hours.  
I have tried SMC reset more than 10 times already.  
I have already tested with another charger.
I have changed the DC-In MagSafe board myself.


The ac adapter shipped with my MacBook is 65w. Will it make any difference to try with a 85w adapter?
Please stop resetting your SMC. You can totally screw up your Mac by resetting it too many times in a row. But yes, it seems through what you tried, your logic board is likely the issue.