I use a Weller SP40NKUK - it's not the best iron in the world, but it comes with a couple of different tip types, including a point one, which is what I used for the joycon mod.
Just try to remember that solder melts very easily when the iron is up to temperature, so be as gentle as you can, you never need to press hard to get it to melt, and the quicker you're done, the lower the chances of heat damage on other components
I didn't use a wire, but I did use some extra solder. The two pins are so close on the rail mod that there's no real need for wire, just enough extra solder to bridge the two pins together.
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So this is why I'm suggesting people go for the solder points on the rail itself. It's an easy solder job and in the worst case you just need to buy a replacement rail, which is like $10 on eBay.
I tried the pin bending and I am not at all convinced it's a good idea. The pins are hard to bend, harder to get to stay in a position you want, and are likely to ruin the pins inside the Switch.
The most professional option would definitely be to solder a little wire between the relevant points on the joycon's motherboard, but that is a much more complex and risky job than the rail points IMHO.
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I'm uploading a video to YouTube of my Fusee Gelee modified joycon happily showing off all of its buttons working, wirelessly. I'm too new to the forum to post a link though, so I'll have to obfuscate the URL somehow.