Don't hold out much hope for getting a static IP working, but please do let us know if it does.
If your system can't communicate with your router (as your ping results show it does not), you can't set a static IP for your computer. Your router's Internet IP is not static; it is assigned by your ISP. But a static IP for your computer takes the router out of the picture of assigning an IP address because the router is no longer a DHCP server for any computers on your system. Your computer will always have the same address on your network, as determined by YOU.
Have you reset your router? If you haven't, turn off the power to it and let it sit about 30 seconds, then turn the power back on. It will probably take a while but your router should go back to its out-of-the-box settings. If you have a separate modem connected to the router (which in turn is connected to your NIC), turn it off, too. CORRECTION: You need to do more than turn off the router to return it to default settings. There may be a Reset button on the router. Check the router's documentation to find out how to return the router to default settings. Turning off the router for a short time is the first thing to do when things aren't working right, but actually returning to router defaults requires more than that.
It's possible to turn off your router's being able to respond to a ping. But if you can't get into the router, you can't check whether it is allowing pings.
By the way, I have the same NIC you have and it works perfectly and has for over six years with every version of VectorLinux that has come out in all that time. I've never had to do anything to get it working. So there has to be some reason it's not working for you. I have no expertise in that area so I'm afraid I can't help you analyze the problem. The driver I use is 8139too. Is the light on the card shining? I've had times when my CAT5 cable has loosened; invariably the light came back on when I replugged the ethernet cable. If the light is not on, the NIC isn't connected to your router and can't work until the light comes back on.
To set up a fixed IP, use VASM, SUPER, NETWORK, NETCONF, then a name and domain for your computer. I use computerbrand.OURLASTNAME (substituting as you wish). This has nothing at all to do with anything outside your computer. In other words, it doesn't need to be a real domain, just anything you decide to call it. Once you've given your computer a name and domain (the part after the dot), choose STATIC on the next screen. Then you need an address for your computer. This is also arbitrary, but you need to have the first three groups of numbers the same as the first three grouops of numbers for your router. ARE YOU ABSOLUTELY SURE YOUR ROUTER'S ADDRESS IS 192.168.1.1? *After* you reset the router by turning it off and doing whatever is needed to restore the factory defaults, if you can't get at it with http://192.168
.1.1 in a browser (if that's the default address), look in your router manual to confirm the default router address (which is also your gateway address). If you are sure you are using the correct address for the router and it's still not working, I strongly suspect a hardware problem. They can fail rapidly. Check with the router provider.
Getting back to VASM, after you assign an address for your computer (the fourth group of numbers can be something between 2 and 255--like maybe 10 or 20), you have to set the netmask. For home or small office networks, that is usually 255.255.255.0 . Then you give the gateway address (the address of your router, 192.168.1.1 if you have it right). Then you give the DNS server address. You should have that from your ISP or you can use something like OpenDNS. That should be it.