Convert NO switch to NC
No transistors need to be abused in the making of your device.  All you need is a resistor:



This causes the full LED current to be drawn all the time, whether it is on or not, but it's really simple and robust.  In fact, the current drawn is slightly higher when the switch is closed and the LED off.

Here is a way to use transistors to minimize the current when the switch is closed:



You can also use a single transistor to get a sortof in between version.  The right answer depends on how important current draw is versus complexity.