I went to Jean-François Duval's tutorial on soldering and milling, and his tutorials were extremely helpful. He explained how to use a solder wick, how to screw the pieces into the modela, and various tricks for SMD soldering.
Milling the board was relatively straightforward. I made a few mistakes initially; I did not properly insert the drill bit, and so the first few times I tried to mill it only scratched the surface. Luckily, after I refit the drill bit the modela milled away
Stuffing the board took about 2 hours, as I didn't really know how to solder before this. After the first few components, it was surprisingly easy. I found that soldering the leads to the attiny individually was an entirely reasonable task
Sadly, when I stuffed this board and tested it on the AVR programmer, it refused to install. I think this may have been because I soldered the USB port badly. I put on too much solder and then did not succeed at using the solder wick to remove it, and may have burned away some of the leads in the process. Thus, I remade the board, this time using the USB Fab Key design, hoping that this wouldn't be an issue:
This board also did not work when I stuffed it. However, I remade the original design one last time and the below board worked:
(Later update) The above board has been a wonderfully trusty programmer and can be seen in several of the other pages :-)