Ok, so I've been able to perform the scan suggested by Ubergerbil. When the computer actually gets an IP address, it pings itself fine, but it can not even ping the router it is supposedly connected to.
Since my last post I have a few more observations. I installed the card on my laptop (Win 7) and it worked perfectly, connecting at 270 Mbps and allowing me to browse the net. I have reattached the card to the Win XP computer and I have a 50% chance of getting a correct IP address using either the Linksys software or the Windows XP wireless configuration utility to connect to the router. When it does connect, it connects at about 13 Mbps (N card to N router) and sometimes spikes to a 270 Mbps connection, even though it reports the signal strength is excellent. When it doesn't connect, I either get message saying Low Connectivity and the IP address is not an appropriate one for the router configuration or it just says "Acquiring IP address" for a couple of minutes and then it says no connection.
Any suggestions on what to try next? Thank you.
edit: Even though it says it was connected and had an appropriate IP address, after 30 minutes it had sent ~300 packets and received 11.