I came up with this after reading http://www.instructables.com/id/An_easier_way_to_solder_magnet_wire_to_SMD_LEDs/
The cost is pretty much zero, and you can do it in under 5 minutes!
Step 1: Get some copper wire
Cut yourself about 10-15cm of wire. Use a knife to remove the insulation.
Step 2: Create your new tip!
When you get to the end, leave a small piece sticking out, and cut it off at an angle. This is your new soldering tip. Cut it shorter than I did in the picture, to reduce the heat loss.
Copper is a good conductor of heat, and so the wire will quickly become hot enough to solder with. Tin it with a dab of solder on the end, and you're ready to go! You can make the point as small as you like by cutting/filing it. If it gets too oxidised, just file it off.