I recorded a video without sound?

Not too long ago I recorded a talk that was given by someone. I used a Nikon D3300 DSLR camera. I inserted an external microphone to the camera to make the audio better than what the DSLR offers as the standard/stock audio.

The problem is that I didn't check that I left the microphone turned off. For this reason the video was being recorded without sound, or in other words the video came muted.

Is it possible to un-mute or add the sound after the initial recording? Or is this altogether impossible?

You can't restore something that was never there.

Some of us who record sound with video, use an external sound recorder to assure high quality audio which we later sync with the crummy sound produced by on camera audio recording systems

If someone else happened to also record the talk, on another device, & you could track that down, you could sync their audio with your video.

By plugging in a mic, you automatically disabled the built-in mic of the camera and since the mic you plugged in wasn't turned on, you are left with a mute video. There's nothing un-mute. No sound to recover. Sorry.

There's no sound on your camera to "un-mute" because it was it was never recorded. You made a rookie mistake. Learn from it & move on.

You can't "un-mute" a recording that was not made!

Always monitor with headphones.

Yes, it's called dubbing.

Use an external sound recorder to assure high quality audio which we later sync with the crummy sound produced by on camera audio recording systems. You could track that down, you could sync their audio with your video.