I had thought that a computer couldn't die if one of the USB peripherals broke
Not true. The coffee probably shorted out the +5V pins in the USB stick. Just like the breaker box in your house, an electrical short can affect everything on that circuit. Every device in your computer connected to the +5V rail from your power supply likely "felt" the short.
If you're lucky, it just damaged the USB port or the wire connecting it to the motherboard. I assume this is a front USB port the device was plugged into. Open up your computer and follow the wire for the port to the header on your motherboard and unplug it (hopefully it's not a ribbon cable for all your front panel lights and buttons). See if the computer boots with it disconnected.
You may have also tripped a fuse in your power supply. Take the cover off it and see if there is any way to reset or replace it. NOTE that even while it's unplugged your PSU can still give you a jolt, so be careful!