The "old router" you mention looks like it is an
ethernet adapter, not an actual router. So that should be plugged into the new router directly with an ethernet cable. It also should not be set to disabled and should be enabled if it is.
EDIT: So my theory so far:
You have one single adapter, a Netgear GA311, meant for wired connections. This should be connected directly to the new router with an ethernet cable. It may be disabled from the sound of things, but it shouldn't be (Right Click -> Enable if so). The IP address it is pulling I'm guessing is static because the default address of your new router is 192.168.10.1. Which means your IP address that you're pulling should be 192.168.10.* and if you give us your sisters IP address, it should look similar. Unless someone manually reconfigured everything, in which case bad on them for not making sure all devices were connecting properly.
So if that's the case, you'll need to go into the properties of the netgear adaptor under Network Connections and go to its properties (Right Click -> Properties). Then from there you will need to select Internet Protocol Version 4 or TCP/IP and select properties depending which windows version you have. Ensure both "Automatic" options are checked and if they aren't, check them. Then close out and try pulling up some pages in your internet browser again.
If this does not work, then we'll want to do several things:
1) Windows Key + R.
2) Type in cmd and hit enter.
3) Type in
ipconfig /release and hit enter.
4) Type in
ipconfig /renew and hit enter.
5) Test internet again and if unsuccessful, continue steps.
6) Repeat steps 1 & 2.
7) Type in
netsh winsock reset catalog and hit enter.
8) Type in
netsh int ip reset C:\resetlog.txt and hit enter.
9) Reboot & try getting out to the internet again.