Use brass wire for the solder, ie, braze traditional lead/tin/whatever soldering isn't going to work out well for steel lampshades. Brass does the job very nicely. You *could use silver solder, but imo brazing is so much a better idea.
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I saw an AC guy use a hanger once to braze with, if you don't want to spend a lot.
Lol...I've welded using hangers. I even used a tire iron as a welding rod once. Crappy welds, of course, but they did the job while they needed to. Gas welding of course.
I would personally go with brazing but I believe silver soldier would work also. You should be able to get brazing rod from a welding supply place. You could probably get away with using a propane torch as long as the metal is thin. Like maybe around the diameter or a pencil lead.
Get a small hobby torch and silver plumbing solder. Do not use lead solder! Make sure you clean the metal with a file and use plenty of flux! Stick on a chunk of solder and heat it with the torch. The molten solder should do the rest of the work.
I suspect the best answer would be brazing rather than soldering. Solder doesn't have much structural strength.