Are you certain that the WiFi card in the laptop supports WPA, some of the earlier models only supported WEP. I have tried a Google search for your model but could not get a definitive answer, perhaps you have a Users Manual which could advise you of the security available.
As "chub_tor" suggests, the Wireless Network Adapter in the Laptop may not support WPA. Updated drivers with WPA support, if available, might resolve the problem, otherwise an option is to plug in a USB WiFi adapter and use that instead of the built-in one.
Thanks to everyone for your responses. Yes, Fruit Bat, our XP's fully updated with SP3 and we're using Windows for the Dell wireless configuration. I see from the Device Manager that a 1394 Net Adapter (whatever that is) is disabled, and that we have a Dell TrueMobile 1400 Dual Band WLAN Mini-PCI Card - when I click 'Update Driver' I'm invited to insert a CD ...that I don't have. Looks like I'll need to follow mgmcc's advice, and get a USB WiFi adapter - a shame, as I've only got two USB ports and they're usually spoken for. Do you have any specific advice on adapters to ask for, or to avoid? Thanks, anyway, for your thoughts.
We’ve stepped back from WPA / WPA2 Mixed to just WPA2 and selected TKIP and got everything working – two generations of Dell laptop, Kodak printer, Samsung Smart TV…
We went back to our settings in response to Secret Squirrel's suggestion - the DrayTek 2800 router wasn't going to reveal its secrets but we had another round of trials at the XP end. Woodchip's suggestion of a PCMCIA card was looking like an excellent fix that would have left both of our USB ports still available, but turns out we've already got there - happy day, and thanks again.