What model is the router?
See here:
http://www.microsoft.com/windowsxp/using/networking/expert/bowman_03july28.mspx

Zxor has it right:
Start, Run, CMD, OK to open a command prompt:
(For Vista or 7 type CMD in the Search box after Start)

Type the following command:

IPCONFIG /ALL

[Note that there is no...

plz do describe fully.

Teamviewer:
http://www.teamviewer.com/index.aspx

Post ipconfig from his machine.

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?

Does your wireless card have a switch? Make sure it's on.
Don't try re-seating the Nic on a laptop. It's not necessary.
Can you connect via ethernet? Is the Wireless Zero service running?

Why would you want to do this?
Broadcasting the signal is handled by the hardware in the access point, and I don't see how a computer can do this. There are many things I don't know, but this sounds...

Do you get an error message?
This problem is typical of some Spyware infections.
Try Malwarebytes' Anti-Malware:
http://www.malwarebytes.org/mbam.php

Sounds like a DNS issue:
ipconfig /flushdns
To copy a cmd Window:
Rt-click top of the window/Select All/Enter to copy/Ctrl-V to paste.

Now we need ipconfig.
http://www.ncsu.edu/resnet/windows/ipconfig/
Post the results here.
Is this DSl or cable?

You shouldn't need to enter anything at all to get this router working.
Reset to defaults.
Connect to modem and computer.
Start surfing.
Your default Gateway should be 192.168.1.1

It looks to me as though the two devices are using identical IP addresses, which is a no-no. (192.168.1.1 is the same as 192.168.1.100)
Change the default address of the router.

See here:
http://www.informit.com/articles/article.aspx?p=1165037
It's long-winded, but it will get the job done.
Networking really sucks in Vista, but persevere and you'll succeed.

Any modern card should do.

What addresses?

There are detailed instructions on the Linksys site. Look up "Using two routers" or something similar.

The DSL Modem/Router is conflicting with the Linksys.
Typically, you have to disable DHCP on the Lnksys and assign it a fixed IP address.

We need a better description of your setup.
It should be:
Modem to router(Wan port)
Router (Ports 1, 2, etc) to each individual PC.

ipconfig?

Wired or wireless?
Post ipconfig from the laptop.

Press and hold the reset button for 10-15 seconds.

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.

Connect your laptop to the router and run ipconfig/all.
Post the result.

Velly interesting.
Check your network connection to see if TCP/IP is installed. Look for the adapter in Device manager.
Run a virus scan.

What OS?
Open your wireless configuration software and select another wireless connection.

That would seem to leave the NIC as the culprit.
Post Ipconfig from this machine.

Does this adapter have its own wireless management software? Look in Programs to see if anything is listed.
Try enabling the Wireless Zero service if it's not started.
Control Panel/Admin...

Get a router. Problem solved.

I suppose there's a reason you don't just use the existing wireless router?
This discussion should help:
http://techrepublic.com.com/5208-6230-0.html?forumID=101&threadID=215676&messageID=2235679

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

It won't add anything, but it won't hurt, especially if it's encrypted.
If you don't use it, turn it off.

This is often a firewall issue.

Toggle the wireless adapter (Fn-2) and try to view again.
If that doesn't work, and your adapter says it's functioning, you need to get in touch with Dell. This has always worked flawlessly for me,...

If this isn't a hardware problem, it might be interference. Microwaves, portable phones and others can cause this.
I'd start by changing the wireless channel, which you can access through the...

That's the icon. I don't think there's any visible sign when you toggle it.
What happens when you right-click?
You should get a list of wireless networks.

Fn-2 is the proper key combo.
Do you see a wireless icon in the system tray?

Run the wireless network wizard.