Why does an UDN2981 lower the current?
I am afraid that all your observations are misleading. 

First, I don't know any white LEDs that would draw 1.7 mA if solid 5V is applied. A normal white lamp should smoke out if you connect it to 5V. 

The reason you have these "strange" reading is that you are using DMM in current mode. The DMM measures currents using a shunt resistor, and therefore there is parasitic voltage called "burden voltage". The burden voltage varies with measurement range, and depends on manufacturer's quality of design. This voltage "eats" you voltage budget, so the LED is exposed to a smaller voltage and hence draws much less current (check with I-V curve for LEDs).

Similar effect happens when you use the UDN2981 drivers, which output is made of Darlington and have the saturation voltage of 1.6V at -100mA, (and maybe 0.6 -1.0 V at smaller load currents, as Jonk mentioned). This voltage drop further reduces the voltage applied to V-I LED curve, which reduces the current to nearly zero.

In general, if you want to measure real current in low-voltage circuitry (under 5V), never use DMM current mode. Use instead a 0.1 Ohm resistor in series with your wires and measure the voltage drop  in mV-mode. Then adjust the value of shunt resistor if your readings don't have satisfactory resolution.