The nut is actually visible on the second pic.
Find a steel M6 (?) nut and ditto matching washer. Screw it for a few cm onto the bolt you used before after you've lubed the threads of the bolt.
Now screw the bolt onto the other side of the frame into the threading of the nut that's stuck in the frame for about at least 1/2 cm deep into that nut.
Once you've done that find yourself a key that matches the nut you screwed on the bolt (should be a flat M6 if that's what the nut was) and start turning the nut clockwise so it approaches the frame.
The washer serves as support area and will avoid scarring the frame. Keep tightening the nut whilst making sure the big bolt doesn't turn along as I assume it will screw right through the frame if you let it.
As you continue to turn the nut against the washer the stuck nut will eventually be extracted out of the frame.
I'm assuming from what you wrote that you ruined the female hex part of the nut that's stuck and not the threading.
If this doesn't work then the only thing left to do is to keep tapping the bolt with a rubber mallet from the brake side, it should become unstuck, eventually.
When the brake will be refitted make sure to use a nut that's long enough, preferably a steel or Ti one, not some cheap alloy thing.
Put a couple of serrated washers over the brake's bolt, insert it into the hole, then fit the nut. By no means turn the brake to level it when the nut is tight already.
Loosen the nut first, adjust and retighten the nut. No need to overtighten it either, that's how it gets pulled too far into the frame and gets stuck in the first place.