Wireless router and a NIC to form a wireless network card

"It is possible, but it depends on your router. Most home routers with stock firmware won't allow you to do this, however many allow you to install custom firmware.

If you have one of these supported devices you can install the DD-WRT firmware. This will allow you to set up a wireless bridge. This allows wired connections to your spare router to connect to your main router as if it were a wireless card.

(Note if you want your spare router to also act as a router for other wireless cards, you need a repeater bridge)"