here is the picture with what you should measure.
You simply need to plug the USB, nothing else
you can use any ground point on the board. the easier one to use is this one. square pad in the connector.
Thanks J4bberwock! I tested the chip as described, and on the non-working blade I do get 5V on the input, but no 3.3V on the output. Just a 0v read.

The working blade shows the correct 5V input and 3.3V output.
I'm assuming this means that the AME8805 is fried. I'm certainly no good at SMD soldering, but do you think I could have this replaced by a pro? Could this failure have caused damage to any of the other components?
Thanks for any other info. Hopefully I can get half my money back from the seller.
This is a "big" smd part, so even with a fine tip iron, you can easily replace it.
The harder thing will probably be to remoove the failed one.
AME8805AEFT (the AEFT thing tells the output voltage and package, so
it needs to be AEFT) can be ordered from aliexpress/alibaba, hard to find for me in usual shops.
datasheet:
http://pdf.datasheetcatalog.com/datasheet/ame/ame8805.pdfIt could be replaced with the following pin to pin compatible, but not tested (it took some time to dig them out)
some of them are available from mouser
AP7215-33YG-13
AP1115BY33
AX1113BF
AX6647-33-E
AX6647-33-F
When ordering, check that these are SOT89 package with the following pinout:
pin 1 is ground
pin 2 is the input (center tab) +5V
pin 3 is the output +3.3v