Yes, it does broadcast as I can get to its security key input.
When I plug the cable in, the lan light on the router doesn't light up - suggesting it doesn't find the PC at the other end of the cable. I have an old W95 PC and the same technique worked fine on that one - I.E. the router saw the W95 PC. I am sure I should be able to configure the PC in a way that the router will see it, but I don't know where to start. I thought the wi-fi solution might be easier. But having seen the next reply below it seems that the wi-fi connection will not work even if I could get connected that way.
Maybe I will start a new question about my network cable!