How to test a Macbook Air battery hardware integrity
There are some apps on the app store that do similar things. Here's a free one I found: Battery Health, but it might not be the best (it was just at the top of the list and looked good).

But guessing from things I heard before, it was probably a part of computer circuitry that was damaged, and most likely the apps won't be able to tell if the problem is really the battery or the computer.

Three other suggestions that might help:

1) Carefully use a volt meter to check the battery (this probably won't work though, because computer batteries are really complicated with lots of different electric pins, and you may end up braking the battery even more).

2) Go to an Apple store and ask them if you can put the battery into one of their computers, or try one of their batteries in your's. If (like me) you don't live near on of those kind of stores, you can try asking around in your local computer repair stores.

3) Gently clean the electronic connections of the battery with a q-tip and 100% isopropyl alcohol. 100% is best, but it's really had to find. So some slightly lower percentages might be alright too. If possible, make sure the battery is completely drained before trying this as it can cause a short circuit if there's power to it. You can also clean your computer's connections with it just to be sure that there is no dried-on gunk preventing contact between the battery and the computer.

Hope this helps :)