Requires mag safe power cord to work after liquid damage.

I haven't had this issue with this model, and I suspect not a whole lot have so you may have difficulty getting an answer, but I'll try to help you as best I can.

The reason you don't get many specific answers for an issue like this is twofold - the subject is very complicated, as you might have guessed, and most people saying "it's this" or "it's that" are going by anecdotal evidence and don't understand the underlying system.

Power management in a modern computer is a complicated system. You have power coming in from the adapter, power coming in from and going out to the battery, power management schemes and charging systems that come into effect, and a plethora of safety systems that vary depending on manufacturer, type of battery chemistry, etc. And that's just touching on a few of the areas that get involved.

The trouble is then compounded by the fact that each manufacturer will take a slightly different approach to all of these issues, with Apple being perhaps the most complex and unique in that they tend to spend a lot of money on engineering their products to push their performance capabilities to the limit (as opposed to those providing typical low-end PCs who can't afford to add this cost to their product and so essentially copy and paste the circuit from the power management controller manufacturer's data sheets). Add on top of that the industry's notorious secrecy with respect to schematics and you have a situation where you often have to reverse-engineer what they're doing.

That being said, the first 2 things I would look to do is go through the console logs for power/battery errors and google the specific error messages to see what others have found to be the culprit, if there exist any comments. Next I'd look at running Apple Hardware Diagnostics and see if it gives an error code that might be able to point you in some direction.

If that doesn't pan out you're going to have to dig into the circuit to find the problem. Set the computer up some way so you have access to the board and can power it as needed. Check the battery voltage. Around 3.2V is usually around 0%, 4.2V is 100%. Does the battery seem to still have charge? Is it in line with the battery meter in OS X? If possible measure the voltage at the terminals with the magsafe adapter plugged in and the mac on. It should be higher than the voltage you got directly from the battery (assuming the battery isn't already fully charged).

All of this would indicate that the charging circuit is working fine and the problem is strictly in the discharge circuit, likely in a blown resistor or similar. If the battery isn't charging then it's likely a problem with the battery itself (it's thermal sensor or charge protection circuitry) or a more serious problem in the power management circuit.

There are going to be testpoints throughout the board that you may be able to identify and use to help diagnose. This is where a schematic (or working specimen to compare to) comes in handy. Frankly, without either it's usually either a very slow or very expensive (fingers-crossed parts swapping) repair.