You need to configure Windows Live Mail to leave messages on the server, so that they can be retrieved again with the second computer.
In the menus, "Tools > Accounts", highlight your account and click the Properties button. In the Properties sheet that opens, select the Advanced tab and tick the box "Leave a copy of messages on the server". Don't leave them on the server indefinitely, otherwise mail retrieval will get progressively slower as the program has to scan through all messages to find which are new messages.
[You may have to press the "Alt" key to display the Menu bar.]
The alternative is to use IMAP instead of POP3. With IMAP, your email program "synchronises" with the server, which continues to host the messages. You then have access to all mail, including your *sent* messages in both computers.