Question Author
Don't know if this message has gone too far for you to read it or not, Ethel, but I've been in touch with Linksys and, after trying various remedies, they have decided that there's a fault with the gateway router and it needs mending with a new one! I'm relieved that this is the case, because I really couldn't believe it could be that difficult to set up a wireless network!
Still, I've learnt about pinging and how to set up a static IP adress as well as assigning DNS servers and subnet masks... it hasn't, therefore, been time wasted!