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
Thanks. However, in my Windows 10 64bits wldcore.dll is installed in C:\Program Files (x86)\Windows Live\Shared. I copied it to C:\Program Files (x86)\Windows Live\Mail and now it works.