Dim LED lamp with constant current power supply using resistors
Figure 1. LED I-V curves.

According to my Figure 1 a typical white LED could be expected to drop about 3.5 V at 37 mA. Four in series would drop 14 V which is in pretty close agreement with your 13.6 V measurement.

Accurate figures for low currents aren't often published but if my graph is any way accurate you should see a voltage drop of about 1.5 V at 1 mA or 6 V across the four in series. You're measuring 12.7 V so your LEDs don't match the graph. They appear to have a much more vertical I/V curve.

Your calculation looks pretty close as far as I can see so I don't know why it doesn't work. 

A more efficient alternate solution would be to connect a series resistor to limit the current. The CC PSU will then rise to 17 V and you'll need to just find the series resistor value as you would with any constant voltage supply.