Because I don't have a Hotmail Account, I just go straight to Windows Live Messenger in the program listing..
- have you tried doing that?
At least you may be able to access the various elements of Live, individually..
It looks like you're running the whole Live suite there, soho..
so there could be a problem with your installation - might be worth repairing it via the Control Panel "Programs and Features" list..
I use LiveMessenger sometimes, but I have MSOffice Outlook to deal with my mail
and consequently do not need a Hotmail Account..
your problem seems to relate to the way in which the various elements of Live interact with one another (ie. a problem with linking, etc..)
Nothing against Hotmail, BTW - MSOutlook just suits my requirements better..
give the repair a try..