The way it's wired A1 is always reading zero (ground).

The 20K resistors are not needed. The analog inputs have nearly infinite resistance so (almost) no current flows through the 20K resistors and there is no voltage drop across them. (Resistors in series can useful as "current limiting resistors" in case you accidently connect more than 5V (or more than Vcc) to an input, you won't fry the Arduino.

The Arduino measures voltage relative to it's ground.

You just need 1 resistor and If you reverse the diode and resistor (with one end of the resistor grounded) you can measure the voltage across the resistor and subtract 5V to get the voltage across the LED. (Voltages sum in series.)

Or the way it's wired now, you're measuring the voltage across the LED so you can subtract to find the voltage across the resistor.

Then, since the current is the same through series components you can use Ohm's Law to calculate the current through the (known) resistor.

Finding the "curve" is a little tricky because you don't have a variable voltage. I suppose you could put a pot in series (between 5V and the LED) and use a 2nd analog connected to the junction of the pot & LED.. Then you can subtract the two analog readings to get voltage across the LED, and again calculate the current (using the voltage across the known-fixed resistor).