The obvious answer to the idle current is to enable standby mode, which I did next. Standby pretty much shuts the processor down, except for the Real Time Clock, which is used to send the wake-up interrupt. When the processor receive the interrupt, The entire boot sequence starts over again, so there is not much use to the infinite while loop in main, because it goes into standby mode before looping a second time. When put into standby mode, the current in the idle loop dropped to 76 uA.

While a low number, 76uA is still much higher than the 250 nA sort of number I am suppose to get with a STM32L0 in standby mode. Pulling some wires, I found that the standby current of the ePaper Display was 74uA, which is pretty high for a device that keeps the display even if there is no current. In fact, just in sleep mode, that 74uA would drain a coin cell in 4 months by itself. 

I tested running the ground through an open drain output, and turning off the ground of the ePD when the device when into idle mode, basically turning off the display. This dropped the current to 2.9 uA. Of that, the SI7091 draws .6 uA, and the rest is from the processor, and leakage elsewhere on the breadboard. Of course I have a PCB for this circuit heading here from China, and it looks like I already have to add a bodge wire.

The code after these modification is tagged StandbyMode and is here.