Fairly sure, unless I missed something. See below.
Are you sure you configured the dns servers on the desktop, as described under "Client set up"?
Actually, one thing important that I just now noticed: the ping on the desktop is actually able to resolve google's IP, it just can't transfer any data.
Below is the output of a few more commands that may be useful. Note: I changed the subnet from 192.168.0 to 192.168.2 so it wouldn't conflict with the wireless subnet.
tcarls@desktop$ grep prepend /etc/dhcp3/dhclient.conf
#prepend domain-name-servers 127.0.0.1;
prepend domain-name-servers 22.214.171.124,126.96.36.199;
tcarls@desktop$ cat /etc/resolv.conf
Kernel IP routing table
Destination Gateway Genmask Flags Metric Ref Use Iface
192.168.2.0 * 255.255.255.0 U 0 0 0 eth0
default laptop 0.0.0.0 UG 0 0 0 eth0
tcarls@desktop$ ifconfig eth0 | grep 'inet addr'
inet addr:192.168.2.20 Bcast:192.168.2.20 Bcast:192.168.2.255 Mask:255.255.255.0
tcarls@desktop$ ping 192.168.2.1
PING 192.168.2.1 (192.168.2.1) 56(84) bytes of data.
64 bytes from 192.168.2.1: icmp_seq=1 ttl=64 time=0.000 ms
--- 192.168.2.1 ping statistics ---
1 packets transmitted, 1 received, 0% packet loss, time 0ms
rtt min/avg/max/mdev = 0.000/0.000/0.000/0.000 ms
tcarls@desktop$ ping www.google.com
PING www.l.google.com (188.8.131.52) 56(84) bytes of data.
--- www.l.google.com ping statistics ---
60 packets transmitted, 0 received, 100% packet loss, time 59007ms
tcarls@laptop$ ifconfig eth0 | grep 'inet addr'
inet addr:192.168.2.1 Bcast:192.168.2.255 Mask:255.255.255.0
tcarls@laptop$ ifconfig wlan0 | grep 'inet addr'
inet addr:192.168.0.104 Bcast:192.168.0.255 Mask:255.255.255.0
tcarls@laptop$ cat /proc/sys/net/ipv4/ip_forward
# I'm using Hardy. The wiki said there's a bug in Gutsy and
# a couple lines need to be added. I wasn't sure if it still
# applied to Hardy to I tried with and without them.
tcarls@laptop$ grep forward /etc/sysctl.conf
# Uncomment the next line to enable packet forwarding for IPv4
# Uncomment the next line to enable packet forwarding for IPv6