How can Kill-a-watt show .1A, 12VA, but 0 watts?
Rounding error on the current reading
Your Kill-a-Watt is defective
Your Kill-a-Watt doesn't read down to 10W
You're feeding a reactive load (transformer, coil, or capacitor) that needs excitation current but doesn't actually consume power (this is a thing with AC)

I'd try it on a known-working 10W load, like a lamp with an LED light in it, or -- if you have one -- a working heater.  If it can't read something that you know is consuming 10W, then it's borked.
To understand the last point, above, AC is different from DC, in that the RMS voltage times the RMS current can be nonzero, while the average power is zero.  This can happen if the voltage and current are perfectly out of phase with each other.  The instantaneous power may flow to the load for half a cycle of the AC wave, but it'll flow back again the next half-cycle.
See the picture for how this can happen: