I had given up on this. The widget automatically picked up (presumably from my system) that I am in Perth so I assumed it would know I was in Perth, Western Australia as that is the setting on my system â€” but the temperatures being returned were ridiculously different to the actual weather here. I assumed this widget was only useful for U.S. users who could also enter their ZIP codes to choose their city.
John Gruber on Daring Fireball explains that by hitting Return, Enter or Tab after entering your location instead of simply clicking “Done”, a list of choices appears and you can actually choose the correct location that applies to you. I tried it immediately and, sure enough, it works.
As John also points out, it’s a particularly bad example of UI design, but now I know the “secret” I am happy. 😀