Sound is generated on the main logic board and converted to analog (what goes to the 3.5mm jack via a DAC (Digital Analog Converter) chip. There is also additional circuity near the jack that handles this audio.
It is likely you fried this. To see if your sound chip still functions, go into the System Profiler and look for the audio adapter. If it shows details, it means it is still functional and only the circuitry at or near the jack has failed.
You could potentially get this repaired, but you’d be better off getting a USB Audio Adapter* or a Bluetooth headphone. It will be less expensive and the quality much better (depending on the quality of the device you purchase).
* If you’re going to connect your Mac to audio gear, my recommendation is to avoid the built in audio. It’s good for the average user, but if you want quality sound, you need quality gear. The DAC in the MBA isn’t that. Besides, if you had a USB adapter, it would have taken the zap and not your Mac (cheap insurance).