Measuring power usage over time? What kit is that?

I think in general people would say to use an Oscilloscope. Take a 1 Ohm resistor and connect it in series with the power, then put the oscilloscope across that.

If you have an oscilloscope that can log, you could export that log and then add up all the values.

I guess an easier method is to get a big capacitor (supercapacitor) and to charge it up to a known voltage. Power Espruino off that, and you should be able to see the voltage lowering over time. The discharge should be more or less linear, so working out the average power draw from that would be pretty easy.