[exim] Sending mail through a smarthost with TLS

I would like to configure Exim to send all of my non-local outgoing mail
through an external relay on port 465. The relay host requires TLS and
SMTP authentication. I do not need to use TLS for incoming mail.

I am confused as to whether I need to set up a private key and certificate
to send outgoing mail (I would assume that I don't need a certificate,
because I can send mail from my mail client via the same relay host
without signing a certificate!).

Currently I have the following in my exim configuration file (edited for
anonymity):

Under "routers":

    send_through_myrelay: 
      driver = manualroute 
      route_list = !+local_domains smtp.myrelay.com 
      transport = myrelay_smtp

Under "transports":

    myrelay_smtp: 
      driver = smtp 
      user = exim 
      hosts = smtp.myrelay.com 
      hosts_require_auth = smtp.myrelay.com 
      hosts_require_tls = smtp.myrelay.com
      port = 465

Under "authenticators":

    myrelay_login: 
      driver = plaintext 
      public_name = LOGIN 
      client_send = : myname : ******

If I try to send mail with this configuration as it stands, I get the
following in my verbose output from exim:

Connecting to smtp.myrelay.com [111.222.333.444]:465 ... connected
waiting for data on socket
selecting on subprocess pipes
selecting on subprocess pipes
selecting on subprocess pipes
selecting on subprocess pipes
selecting on subprocess pipes
ok=0 send_quit=0 send_rset=1 continue_more=0 yield=1 first_address is
not NULL LOG: MAIN
SMTP timeout while connected to smtp.myrelay.com [111.222.333.444] after
initial connection: Connection timed out

What should I do to get my TLS and authentication working?

thanks,
Matthew.