Re-radiating GPS isn't a great idea, I'm surprised someone has tried it in that link you posted. The signal power is so low (~ -150dB), you can easily swamp it and degrade other GNSS receivers in the area if you take a 29dB amp and 'broadcast' the signal. Not a great idea in aviation. Been there, done that, never doing it again.
You can get USB GNSS dongles with a connector for an external antenna. You would need to check the datasheet for the antenna to see what power it needs. I can't find one after a quick search, so how old is it? It may have failed if it's not been connected to anything for a while, so you'd need to test it. Those take between 4.5v - 15v I think, depending on the actual part number (there are several AT575 variants). A USB dongle will provide a max of 5v DC bias and may be happier with a higher voltage.
You're making your own GPS Mouse with an external antenna.