Tray Icon Says Not Connected But Web Works

Some time ago the mother-board ethernet card on my Dell Dimension 8400 failed, so I fitted a PCI card, and all was well: The failed card was Broadcom NetExtreme 57xx. The PCI one is Realtek RL8139. I got back onto the web, even though one of the (now two icons) in the system tray has a red cross, and all seemed well.

Recently I wanted to set up a home network with a wireless laptop, and though both machines seem to see each other I cannot connect the PC to the LAN to share files. The Ethernet icon in the tray has a red cross, and gives the pop-up "Not Connected - no connections are available". The Network and Sharing Centre will not let me connect, apparently because it thinks I am connected to a public network, which I cannot change to work or home. but it also has a red cross on the PC to Internet connection. If I run the troubleshooter, it cannot find and fix any problems. In the security settings app, I have set the radio buttons to be able to change the status of networks, and to rename them. If I try to connect from the laptop to the PC, I am asked for a network ID and password, but there is nowhere for me to set this up. I have a wireless dongle, but this gives the same symptoms. I am able to connect to a wireless printer (ethernet cable to router, then wireless to the printer).

I have tried many things, always making sure I have Administrator privileges. I don't know whether the problem is that the PC only sees the faulty modem, or is something to do with the public network status, which I have seen as a problem in many other posts, but none of the solutions have worked. I have uninstalled the faulty modem, but obviously cannot physically remove it, as it is part of the motherboard.

System information is thus:

Tech Support Guy System Info Utility version 1.0.0.2
OS Version: Microsoft Windows 7 Professional, 32 bit
Processor: Intel(R) Pentium(R) 4 CPU 3.00GHz, x86 Family 15 Model 3 Stepping 4
Processor Count: 1
RAM: 3070 Mb
Graphics Card: Radeon X300/X550/X1050 Series Secondary (Microsoft Corporation - WDDM), 4 Mb
Hard Drives: C: Total - 152335 MB, Free - 110764 MB; E: Total - 324599 MB, Free - 268186 MB;
Motherboard: Dell Inc., 0J3492
Antivirus: Virgin Media Security Anti-Virus, Updated and Enabled