The PCB for the crazy tiny keyboard arrived as well, and I had a stab at soldering the buttons by hand. I got as far as the second button of the second row and decided that it's not going to work. My tiny soldering iron tip is simply too big for this.
I might give it another try with solder paste and hot air gun, but I really don't have much hope for it. It's also pretty inconvenient for typing with your thumbs, so I guess I will just call this one a failure.