Dell wireless client vs linksys router

First, I would point you to the doc on WiFi for that particular laptop : http://support.dell.com/support/edocs/systems/latd610/en/ug_en/wireless.htm#wp1134874

Next there are some simple things that are often overlooked.  I would check to see if there is any physical switch or button that would toggle the on board WiFi on/off and set it to on.  My sales guys always forget this.

Encryption standards and keys are an obvious consideration if they have been selected.  I'm sure you know to match them.  

I can not tell if you can see the SSID or not by your description so I have noticed that when initially connecting to a new router if the SSID is hidden, even if you specify what it is on the client, it will not find it.  To correct this I just set the router to broadcast and sniff it out and connect then after the connection is made I hide the SSID again.  This may not be reliable, and require this method over again as needed.  

Make sure there is no MAC authentication list.  If this is a new device you would have to authorize the new MAC address on the router.  Though in this scenario you should still see the SSID even though you can't connect.

I would also make sure the router and the intergrated wireless nic are using the same bands; B, G or whatever.  Sometimes the routers can be configured for B only, G only or both B and G.

I would work backwards on the router setting it to the easiest access with no encryption, no filters, no hidden SSID, allow DHCP.  And see if you can connect in this manner.  Then turning on security one by one and seeing if the connection still binds.

If you still can't connect with the easiest settings then are you are able to connect to any other wireless access points to determine if it is actualy the Dell or not.  You may try bringing it to a Hotel, Library or Coffee House where they have free WiFi and try to connect there.  If you can connect there it may be the router that is getting goofy on you and need to factory reset and reconfigure.  Although I would only do this as a last resort because it's such a pain to reconfig sometimes.  I've seen stranger things happen and have done this before.  

I have also seen problems with a Dell laptop last weekend trying to connect to a Linksys BEFW11S4 though the laptop was running Vista.  If yours is the same there may be issues with that unit and out of box WiFi config from Dell.  

If the intergrated WiFi is a bit buggy on the Dell then you are on the right track with selecting a PCMCIA solution here.