" I have 6 computers connected through a switch (I think one of them does the routing... not sure) for a taxi booking system. I see no router. None of them have an internet connection."
If there is no internet access, there would be no need for connection to a router and I cannot see why these computers would have a "Default Gateway" address. They need only be connected to the Network Switch and given fixed IP and Subnet Mask (not Subnet) addresses. As this is just an internal "intranet", try deleting the Default Gateway address in case that is causing the problem with the PC that you do want to give internet access to.
"I've added a USB to network adapter as a second NIC on the PC I need to get internet onto and assigned a static IP address to it (10.10.0.20)."
This adapter will need to have:
- An IP address. 10.10.0.20 is fine provided the router is operating in this subnet (IP address range) and it doesn't conflict with any other devices.
- Subnet Mask address. Should be 255.0.0.0 with 10.x.x.x IP addresses, but using 255.255.255.0 won't matter.
- Default Gateway address. This must be the router's own IP address.
- DNS Server addresses. Virgin Media's DNS addresses are 18.104.22.168 & 22.214.171.124