It was running an old version of Linux so I updated it to Ubuntu using the process to upgrade an existing Linux installation over a network connection. Worked like a dream.
At the end of the process I rebooted the machine but then realised that Ubuntu doesn’t run a ssh server by default.
I couldn’t log in on the console because I had no monitor and the default Ubuntu install boots to the Gnome GUI which is impossible to drive ‘blind’. Here’s how I sorted it.

1. Open one of the alternate text terminals by hitting ctrl-alt-F2 on the machine’s keyboard.
2. Enter your username, press return.
3. Enter your password, press return.
4. Enter sudo apt-get install openssh-server, press return.
5. Enter your password, press return.

With careful typing, that should get you logged in and install and start the ssh daemon. You can then log in remotely.