Steve, I had the same problem on my 58. The welded nut had broken loose from the back of the panel and the bolt and nut just spun as an assembly. I took it apart just the way jockbolter suggests, by wedging a prybar against the nut to jam it while I turned the bolt out. Not alot of room to work, so you'll have to get creative with what you use for a prybar and the angle of attack.
It wasn't easy, and I think I invented some new cuss words, but I eventually got it out. If the nut and bolt have grown together over the past 50 years you might want to soak it good with penetrant for awhile first.
Putting it back together was another story. It's been awhile since I've done it, but I don't believe there's any way to get to the back of the panel to weld a nut back on. My solution was to take a slightly longer bolt with a new nut and lock washer. I reversed the removal method with the prybar. Once the lock washer started to grip it was fairly easy to tighten everything up.
Somebody correct me if I'm wrong, but I think the only way to weld the nut back would be to either remove the fender and cut a hole in the back of the hinge pocket from the lower cab/firewall side to get at it, or buy one of the patch panels with the nuts already attached from one of the venders, cut out the old section, and weld the new one in.
If I were to make the same repair today, I'd be tempted to try removing the hinge assembly and using epoxy to "weld" the nut back in place. I'd put the epoxy on the nut, and put the nut and bolt together tightly in the bolt hole to keep everything aligned while the epoxy cured. Once it set, in theory the nut should stay securely in place after you remove the bolt. If you try this, I'd recommend using a putty type epoxy instead of a liquid so you don't accidentally get it on the threads and glue the nut and bolt up tight. Of course, if the metal is rusty on the back of the panel this fix might not work too well.
Just FYI, the hinge repair kit I bought came with hinge pins that were about an inch too long. I ended up using the new bushings and the old pins. The new pins would have worked, except the hinge assembly won't fit back through the hole in the pillar with the extra pin length hanging down. This was about 9 or 10 years ago, so hopefully that problem has been corrected by now.
Bobby