Here's what I did based on
a) @Espo 's comments and
b) the fact that I only had Windows Task Manager to play with....
I logged onto the host machine, opened Task Manager and used the view menu to add the PID column to the Processes tab.
I wrote down (yes, with paper and a pen) the PID's for each and every instance of the vmware-wmx.exe process that was running on the box.
Using the VMWare console, I suspended the errant virtual machine.
When I resumed it, I could then identify the vmware-vmx process that corresponded to my machine and could kill it.
There doesn't seem to have been any ill effects so far.