Most tips on modern irons are electroplated with a combination of iron, nickel and chrome, so you can't grind them down unfortunately.
You may be able to get a replacement tip for your iron which is smaller, otherwise you'll have to track down a better iron. Highly recommend the Goot iron I mentioned above, it makes life very easy. Your 15watt iron will still be handy for power and ground points, the larger wire is a bit easier to solder if you have more power.
A couple other solutions for soldering legs on IC's I've heard mentioned are coating nearby legs with small amounts of petroleum jelly so solder wont stick or covering them with masking tape. No idea how well they work, might be worth experimenting with on an old computer board.
PSTwo V14 SCPH-75002a: Crystal Chip 1.1
XBox 1.6b: SmartXX v2.0