You may have to disable the NIC, but try to change all the settings in Network Center first.
Change "Private" to "Public ", then back, etc.

Vista is sometimes very stupid about networking.
Delete all the networking entries on one machine and set it yp again.

What OS?
Is there a share on the laptop?

VNC will do it:
http://www.realvnc.com/
It's quite an education to set it up, but it's used by a lot of people.

I believe both routers must support WDS, and it looks as though the Netgear may not.

Listen to jbennet, as he's got it right. A router makes this exercise a piece of cake. I agree with the Linksys recommendation as well.
Forget the switch-you won't need it.
NAT routers have a...

Would someone please tell me what the "Windows Outreach Team"
is?
They seem to be posting a lot, and often with bad advice.

A fixed IP address is used for certain situations like port forwarding, and it sounds as though you don't need it.
To access ipconfig:
Start/Run/cmd
Type ipconfig at the command prompt.

There is no need to install any software anywhere. The router needs to be properly configured, and you need a connection set up on each computer, but all the software you need is native to windows.

You can get some help with ICS here:
http://www.homenethelp.com/ics/ics-install-arch.asp
This would be significantly easier with a router.

I wouldn't think so, but crazier things have happened.
Post ipconfig /all from this machine.