Let's proceed step-by-step:
1) Could you delete windows.old with disk cleanup ?
2)
but the fundamental problem of the Administrator password remains. How on earth can there be a password when I didn't set one, and how can I fix it ?
Have you tried to take ownership of windows.old, right-click the Windows.old folder, select Properties > Security Tab > Advanced at the bottom > Owner Tab > Edit > Highlight your username and put a tick in ‘Replace owner on subcontainers…’ and Apply > OK. ?