MacBook Pro 13 Early 2011 Battery Degradation to Not Detected, User Replaced New Battery also Not Detected, Please Help!

I have a MBP early 2011 13 inch and do not have Apple Care. After I installed Mountain Lion I noticed the battery health was dropping pretty rapidly, with a service battery warning. I tried resetting the SMC and PRAM and even a clean install. I took it into an Apple store and they said I should have cycled it more, but I could still use it. The health dropped and dropped until a battery would not even be detected. Because Apple would not replace it for me for free and I didn't want to pay for them to do so, I bought my own battery. Once I replaced the battery with the new one, it still says there is no battery connected.

When I detach the battery and then attach it again, the battery indicator lights on the side show 4 dots. When I plugged in the MagSafe connector, the fifth light started blinking (I thought it was charging) and then it went back to all of them blinking indicating no battery detected. The computer also automatically turned on when I connected the MagSafe.

I have tired multiple times to reset PRAM and SMC even after installing the new battery to no avail. I can use the machine with the MagSafe connected, but as soon as the cord is unplugged or bumped, it goes into the emergency sleep mode.

Any thoughts on what might be ailing this not yet two year old machine? My first Mac lasted 7 years before any logic board fail so I hope that is not it. Anything I should try that I haven't yet? Any ideas on what this might cost me through Apple?

Thanks!