Aluminum MacBook Charging Issue

I have an old MacBook Aluminum (Late 2008) that I have recently returned to service. It was just a machine I would occasionally use when I was too lazy to break out my work MacBook Pro. It had been up for months with no issues running Snow Leopard (10.6.8) when I decided I wanted it to be less painful to use.

I opted to upgrade it to El Capitan (it is running 10.11.4), then did a full clean install of El Capitan (there was nothing on there I didn't have elsewhere). Everything went super smooth. I've been using the machine for a few weeks and just recently ran into issues with charging.

The Symptoms:

The charge indicator on the side of the body flashes the final light when pressed. The OS reports 99% charge and charging. The power adapter stays orange. It stays like this indefinitely until I take action.

Temporary Workaround:

If I use the machine and drain the battery below ~70%, plug her in, close the lid, and leave her be she will hit "99%" and the light will turn green. However, if I unplug her for any reason and she goes to battery, the next time I plug her in unless I've significantly drained the battery she will charge indefinitely.

Troubleshooting Thus Far:

I have done an SMC reset and the workaround mentioned above. I was unable to run a hardware test from boot; I think I need the original Leopard install CD and I'm not even sure I have it at this point. I might have a copy of Snow Leopard lying around though if I really need it.

Other Info:

I'm pretty sure the battery has bad cells; she clocks out at about 20-30% charge (hard death). A few years ago she was dying at 89%; the Apple Store determined the battery had dead cells and had to be replaced. I replaced the battery with a genuine Apple replacement provided by Genius Bar. I currently have no aftermarket hardware in the machine (though I was toying with the idea of throwing a Crucial SSD in there, provided I can solve this charging issue).

Thanks.