"If I type explorer.exe into task manager it comes up with not found"
That's why you've got no icons/desktop then, probably. Explorer is what runs that.
A registry key that handles how the Windows shell (explorer.exe) loads is likely corrupt. Use Task Manager (CTRL+ALT+DEL) to run regedit manually. Then find the following key and delete it:
Image File Execution Options\explorer.exe
Then reboot. This is a serious error - may be caused by a virus, so check your system out very carefully.