How do I get a wireless PC internet connection with no wifi ability?  

When I come across an otherwise good Notebook where the Wi-Fi has failed I use a USB Dongle, favorite is this one:
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There's many more on Amazon.com. The dongle can be as small as the one used by a Wireless mouse. There's also Add-in cards, either PCI or the newer PCIe [PCI Express] depending upon what a Desktop can handle.

Alicia J said:

Does the WiFi usb dongle have to be used in a pc that is in the same room as the modem?

Generally No but there are other considerations such as what will block radio signals, what kind of insulation is in the walls, with older buildings was lead-based paint used, are there power-hungry appliances between the two [electricity generates a flux around such things and the power cables that can also interfere but that can also interfere with Ethernet cables]?

Alicia J said:

Would one with an antenna be less likely to be an issue? When I moved the modem to the other room, my cell still had a strong wifi signal in the room with the PC. There's no lead paint in this house, and no heavy electrical appliances have interfered when the PC was hard wired (ethernet).

Yes, an antenna can help. Example is the newer Routers that are available with 2, 3 or 4 antennae on them but they are pricier and to get the best performance the Wireless adapter in the computer/s need to be the same as the Router, either 802.11n or 802.11ac.