Can anything be done to get the full range?
No not with a variable resistor, you really need a pot.
I imagine 'analog input' is the destination of the current on the arduino board and 'input' is the +5?
No, input is the analogue input. You need a resistor between that and +5V.
THEN your two wire thing goes between the analogue input and the GROUND.
Finally the capacitor is wired between the two connections of your two wire thing.
The significance of the LED I do not understand at all
The LED is acting as a pull down, if you wire it as I say you will not need it.