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The first thing is that you probably want to use SOLDERABLE magnet wire (such as that coated with self-fluxing polyurethane insulative coat). You don't have to strip the wire before you solder it which makes it easier to solder small connections and the insulation that remains wherever your iron didn't touch prevents shorts. But take note: the fumes produced from soldering through this type of insulation are REALLY bad for you. We're not talking about regular solder fumes here. Definitely have open windows and a fan blowing. I'd try 30AWG magnet wire. It gets hard to work with anything thinner.
Why don't you pull out a metric ruler with mm and put it up against the connector and measure the pitch spacing? If it's too small, measure a few pins and divide up the measurement by that many pins. Then look up a table of wire AWG diameters.