Let's assume your sink has resistance to limit LED current
Your main problem is that your arduino input is varying from +12volts (LED off) to roughly +10volts (LED on) (based of a forward voltage of 2 volts)
To read these values you will need a potential divider to limit the arduino input to 5 volts (or 3.3)
So now you are looking at 5 volts (off) to around 4.2 volts on)
Whilst this is "do-able" it will be subject to the stability of the 12 volt supply and the forward voltage drop of the LEDs. With lower forward voltages it will be even harder to discriminate reliably between off and on state.