Can I solder a resistor into the wiring to dim the LED substantially or should I just go out and get a dimmer LED to replace it? What type of resistor should I use?

Sure you can. Exact value of resistor depends on supply voltage (likely to be 5 or 12V), the forward voltage drop of the LED and wanted current through the LED - the last two factors are open for speculation as we don't know the exact type of LED.

Therefore my suggestion is to use a small potentiometer (variable resistor) of 4.7k or 10k Ohm value. Wire one of the side legs together with the centre leg. Snip one of the wires to the LED and connect one end to the single leg and the other to the combined legs. This allows you to adjust the resistance from 0 Ohm (ie same state you have today) up to the value of the potentiometer.

/Michael