My battery isn't detected and won't charge

I have a mid 2013 MacBook Air (MacBookAir6,2) and have had trouble with charging and now with a new battery not being detected.

For quite a while (probably more than a year) my laptop has had a quirk where it will only charge if the MagSafe connector/cable/laptop are on a particular angle (i.e. the cable needs to be higher than the laptop, or the laptop has to be on a slant with the MagSafe side higher). This was annoying, but I could still reliably charge my laptop without too much hassle.

Last week, however, my battery stopped charging altogether and the light on my (third party) MagSafe charger disappeared. The situation, then, was that I could boot and use the laptop while the power adapter was connected, but the charge level would never go above 0% and as soon as the power adapter was disconnected, the laptop would shutdown. With the cycle count of the (original, factory-fitted) battery at just under 1750, Google seemed to suggest that it was probably worth buying a new one.

I installed the new battery (a NewerTech NuPower 55 Watt-Hour Battery) last night, but nothing has improved. In fact, the battery status is no longer being shown in the menu bar (when I tick that box in the Energy Saver section of System Preferences, it immediately unticks itself), and neither the system report or coconutBattery seem to be able to detect the battery at all.

I have reset the PRAM and SMC many times (with both new and old batteries), disconnected and reconnected (what I believe was) the LED indicator light module, and am now completely out of ideas.

I would be extremely grateful for any advice.