Batch file is the answer.
Assuming u: drive is the dead PC's drive that's attached to the current PC and the user profile is admin.
Code:
echo d | xcopy u:\users\admin\Desktop /Y/F/S c:\users\admin\Desktop
echo d | xcopy u:\users\admin\Documents /Y/F/S c:\users\admin\Documents
echo d | xcopy u:\users\admin\Downloads /Y/F/S c:\users\admin\Downloads
echo d | xcopy u:\users\admin\Favorites /Y/F/S c:\users\admin\Favorites
echo d | xcopy u:\users\admin\Music /Y/F/S c:\users\admin\Music
echo d | xcopy u:\users\admin\Pictures /Y/F/S c:\users\admin\Pictures
echo d | xcopy u:\users\admin\Videos /Y/F/S c:\users\admin\Videos
the reason using echo d is because xcopy will ask whether the destination is a file or a directory and user should answer (d)irectory.
/Y Suppresses prompting to confirm you want to overwrite an existing destination file.
/F Displays full source and destination file names while copying.
/S Copies directories and subdirectories except empty ones. (sorry, missed this one earlier)
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/Q Does not display file names while copying.