This is weird...
I got it to work, I don't know how. Then I restarted windows without touching the DMM or the serial adapter and now it doesn't work anymore.
This looks like it's either a bad solder joint (but I don't think so as I didn't touch anything) or a windows/driver problem.
I have major software problems in connecting a OBD scanner for my car. It's USB to Serial, and shows as a COMx in Device Manager, but doesn't really work. As it turns out, Toshiba Bluetooth Stack keeps screwing up the COM ports, somehow. I made it work at some point, but I don't remember how. I used it for some time, all was good, worked every single time. Lost it after a restart (weeks later, because the laptop always went into standby). It's frustrating because I can't remember what the flick I did.
I suspect you might have the same software issues.
I will dig more into how COM ports are managed on a modern PC and possible causes for other people's (unrelated) equipment not working on serial-over-USB.