Maxbert: clearly the PC is seeing the drumkit arrive and leave - the sounds tell you that. It's something "downstream" that's misbehaving, either the drivers or the drum software.
Some of this stuff is fussy about what order you install things in - software first, or driver first, and whether you should install the software before you plug in for the first time. Check the instructions carefully to see what order they tell you to use.
What worked for me on a different bit of kit was to remove completely, then reinstall in accordance with the instructions (I know, I know!).
- Unplug USB device
- Control Panel, Device Manager, Remove (or Uninstall) device
- Control Panel, Add/Remove Programs (or "Programs and Features" if that's what it shows) and remove the drumkit software.
- Power off the machine (nost just "restart")
- Power on and follow the destructions.