Problem partially resolved
You're right, this really is a cheaply made converter. When I plugged the cable into the digital input, I unwittingly pushed the input receptacle out of the board it was plugged into. I took the cover off the converter and reinserted the pins into the board (there's practically nothing to keep them in place). Then I plugged the cable in, being very careful not to unseat the receptacle again.
Now I'm getting sound through the external speakers, but only on TV programs. I tried a streaming movie from Netflix, and there's no sound, just a little intermittent static. Maybe that's the 5.1 audio you referred to?
In any case, I'll need to get a different converter. I'll contact LG support and see if they can recommend one.