Basically from phone line to computer you need the following:
Filter. Just a little box that plugs into the phone line and prevents interference between the data and voice parts of the line. Literally a couple of £s, or $s.
Modem and wireless router or Wireless modem/router. As say your daughter already has a modem, just a wireless router would suffice. However, as there will be a need to purchase a router anyway, to get one with an integrated modem would probably be worthwhile just for simplicity and convenience. Over the cost of a stand alone router the extra cost is minimal and its one less thing to connect up, find a power supply for and have wires trailing between. £70 (or equivalent) for a reasonable wireless modem/router. Probably £40+ for just the wireless router if you decided to use the old modem.
PCMCIA Wireless adaptor. Basically the 'mate' of the wireless router that goes in the laptop. As to whether the laptop is already wireless enabled, being 18 months old unless it was a really top spec one, I'd have thought it unlikely. A search of the exact model number on the net will yield the full specs to be sure however. £35 or so over here.
Thats all the kit required. Many places sell bundles. A wireless modem/router, PCMCIA card to match and any other bits and bobs required. These often represent good value for money. It's well worth a shop around the various computer bit retailers websites to find a good deal.
In addition, I suggest asking opinion of the kit you're thinking of selecting from those here knowledgeable enough to know the good from the bad as, I'm told, some brands are trickier to setup/more tempremental than others.
As for setting up. I was lucky perhaps, but I pretty much stuck the PCMCIA card in my laptop and let windows install it, plugged my modem/router into the filter in the phone wall socket, then turned it all on, went into the setup screen of the modem and gave it my ADSL account details and it all worked. I did later have someone explain to me how to secure the network as I had no idea of the best way of doing so.
I appreciate that the setup/securing bit is going to vary from modem to modem to some degree so trying to give detailed explanations is probably futile. Even having someone who's reasonably familiar with setting up such things talk through it on the phone step by step as she goes should be sufficient to see it done though.