After spill: needs to be connected to charger to simply turn on

I spilled a bit of water onto my keyboard a little less than a year ago. After the incident I flipped over the Mac and let it dry for a week or so.

Everything worked except for the battery. My Mac simply will never turn on unless the charger is connected to it. If the charger falls out, the Mac shuts off.

Now I've managed to work like this for so long, but my other problem is the battery does not actually recharge, so its been going from 100% and gradually dropping. Now its at 10% and it can only decrease so I am wondering how I can fix it.

I've been to Apple, those cons want to charge me over $700 just to replace the battery, they say its connected to the trackpad or something as such. They did not open it. I also went to MicroCenter and they gave me a quote of "several hundred" but never given an exact amount. They also did not open my computer.

How can I fix this on my own? I've seen mid 2013 videos for battery replacements and the battery doesn't seem to be connected to the trackpad or keyboard or anything.