Step 26: The hard part
First get your flux. As I said, this is pretty much required. I can't imagine making a good connection here without it. I am using zinc chloride flux, which is acid based. This is not recommended for electronics, because it is strong enough to erode the trace. BUT, as long as you use a small dab and don't heat the connection too long, you will be fine. The reaction that eats away the copper is greatly accelerated by high temperature. Once you remove the heat, it might take a couple of lifetimes for the flux to eat away the trace on it's own. Plus, once it dries out (it's a hydrophillic suspension) the reaction will slow even further.
See picture 2: Now get a lil bead of solder onto the end of your 30AWG wrapping wire... You must use 30AWG wire, as the tip will fit inside the indentation of the via, providing capillary suction to the solder.
Once you have a bead as in my picture, cut the end of the wire off so that just a teeny bit is poking out of the bead.
Dip the bead into the flux. You don't want to see a visible gob. Just know that you did, indeed, make good contact between the bead and the flux, and that should be enough.