But I felt like an idiot when I tried if for the first time and repeated "Hey Siri's" fell on deaf ears. So much for impressing the wife.
Duh. The "Hey Siri" feature has to be configured; it's not turned on by default.
Go into Settings->General->Siri. Then turn on Hey Siri:
You then go through a few screens so Siri learns your voice and you're good. Yep. Works a little better if you turn it on...