[Al Skierkiewicz]

Nathan,
What you are trying to do is one of the most difficult soldering jobs there is. From the picture, I will assume you are trying to solder to a connector of some sort. This will prove to be bad for the connector is not done correctly. The first thing you have to realize is that an iron no matter how hot, must conduct the heat to the work. If the tip is dirty or corroded, no heat will transfer. Start by touching the tip with some solder to form a pool of conducting solder to transfer the heat. If you need it, a wet sponge can be used to clean the gunk off the tip.
Second, the wire must be clean to accept the solder once heated, Sandpaper works OK for this, be sure the wire is nice and shiny. Then tin the parts separately by heating and applying solder to the metal. Note in your picture how there is solder on the wire almost 1/4" from the end? That came from the tinning process. Do the same for the connector leaving enough solder to form a blob on each of the intended targets on the connector and wire end. When the wire is hot enough to melt solder then bring all three things into contact, the iron, the connector and the wire. When you see the solder start to flow, remove the iron and hold the wire in place while blowing on it to cool it down. You might want to try soldering scrap wires together before tackling the connector. There is no coming back from a damaged connector at the frequency this antenna is designed for. A good connection will have a shiny appearance on the solder surface. If it looks textured or crystalized, it is cold soldered and needs to be redone. As others have pointed out, this type of connection is only as sound as the solder so don't expect it to hold together with movement. Let us know how it goes.
BTW, if you hold the wire with a pair of pliers near the point you are trying to solder, or if you have the wire in a vise to hold it, the heat may be drawn away from the tip of the wire. Try holding the wire further away. I think when you add some solder to a clean iron tip will you will have much better results.
I forgot to address the suggestions about the microwave. Please don't do this! Damage to you and the microwave may result.