Have you tried to uninstall the Logitech Cordless Trackman via the Device Manager node of the Computer Management console?
I would try to uninstall the device there and then run a scan for hardware changes to see if Windows can automatically reinstall the hardware to working condition.
To reach the Computer Management console, you can do the following:
Start >> right-click My Computer >> select Manage -OR- Start >> Control Panel >> Administrative Tools >> Computer Management
Navigate to the Device Manager node. Locate the Logitech Cordless Trackman under the Mice and other pointing devices branch. Right click the device and select Uninstall. Once the uninstallation process has completed, right click the Local Machine at the top of the hardware tree and select Scan for hardware changes. This should start the Found New Hardware Wizard. Try reinstalling the hardware that way.
Try to restart your computer after uninstalling, when your computer reboots, the Found New Hardware wizard should start up and attempt to install the Logitech Cordless Trackman.
I would start there. Post back for further advice if that doesn't resolve the issue.