Amplifying signal with Darlington not working
Your problem is that you have ignored the part of the data sheet which states that the sensor output is "open collector NPN". Disconnect the Darlington, and connect a 10k resistor from the sensor output to +5. The fact that you're seeing 220 mV is just an artifact of leakage. 

With a 10k to +5, you'll see 0 - 5 volts pulses, and you can tie these directly to your Arduino.

And, just to make things clear, you must also get rid of your pulldown resistor.