Somehow deleted the whole Deleted Items folder from Windows Live Mail.  Thus, deleted mail disappearing entirely.  Tried creating a new "Deleted Items" folder.  This didn't work either.  

Finally, discovered the following to restore it:

Under "Tools",  select  "IMAP folders".   Then select "Reset List"  (far right hand side).  Then click "OK".

That's it.  The folder instantly reappeared and is in working order!

Hope this can help someone with the same problem.