Then Right-Click Database Mail/View Database Mail Log and see if there's an error that holds clues. In my expereince, when I can't send a test message it's one of 3 things...
1) Something wrong in the Database Mail Account configuration. Check with your email server person to make sure the server name, Port Number, SSL and SMTP Authentication settings are correct.
2) Anti-virus. Check anti virus software for settings that prevent the server from being hijacked to send spam.
3) Email server security. Ask your email server person to look for anti-spam security settings that prevent the database server from connecting to the email server. I had to go through this process with my mail server guy mutliple times to enable all of our SQL instances to send mail.
Once you can send a test message, xp_sendmail will work if you have the right parameters.