I had same issue. Here is how to fix it. This is easy and works! Thanks to whoever figured it out.
After long day of research, at last I got the solution. It seems the new Windows Live has grouped the
wldcore.dll and wldlog.dll (used by Messenger, Contacts and Mail) to a folder
c:\Program Files\Windows Live\Shared

Maybe because I did upgrade install it screwed up things.
The solution was simply copy paste those 2 files wldcore.dll and
wldlog.dll from c:\Program Files\Windows Live\Shared to
- c:\Program Files\Windows Live\Messenger
- c:\Program Files\Windows Live\Contacts
- c:\Program Files\Windows Live\Mail
Now I can login to Windows Live Messenger and enjoy the whole Windows Live Essentials 2011 products.
How do I paste to the c\Program Files\Mail? Where do I find these programs?