LED circuit analysis with PHY DP83822HF
how can I figure out what amount of current (voltage) is going through each leg of the circuit?


I'm assuming you mean when the output is low.

Left one is easy...it's 1.8/1960 &lt; 0.918mA

Right one is more complicated but will be a lot less than that and certainly not enough to light the LED like you want it to. See next..


  Will the LED have enough Forward current to turn on (20mA). 


Looks like those outputs are only rated for 4mA, and you stole 1mA with the other resistor, so even with no resistor above the LED, at 3mA it will be very dim at best. With an added ~2K resistor above the LED, you do not have a hope.

Further, if the VF of the LED is greater than about 1.7V it will also be significantly dimmed.

In order to drive an LED you will need a higher voltage. You may already have one, or you can add a boost circuit specifically designed for this kind of purpose.