Awesome tutorial! I had never thought about using desoldering braid as a conductor on fabric!
I think you may have gotten the math a little off, though. You can include the voltage drop of the LED when calculating the resistor value. So, if your LED has a 2V drop and 15mA draw, if you're feeding it with 3V, the resistor would be (3V - 2V)/0.015A = 66.67 ohms.
It's not a big deal necessarily, but if you don't factor in that voltage drop then your resistor will be more ohms than it needs to be, which means the LED will be significantly dimmer.
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