The nginx setup is free of any costs, but if you live in the same house an easier way would be using a capture card in your PC. Your fiance's PC just has to clone his desktop output, for example using the hdmi or dvi port of his graphics card, and connect that to the Capture Card in your PC. Now in OBS you can add the capture card to any scene you want it on, and it will show whatever your fiance does on his cloned monitor :)
Good capture cards would probably be the Avermedia C127 Game Broadcaster HD or the C985 Live Gamer HD, also from Avermedia. Both work very good with OBS and are not the most expensive. Also both allow input up to full 1080p (which can be recorded at 30fps). They should be in the range of 100$.
The next "better" capture card would probably be the x-capture which offers some more inputs and features but costs around 200-250$ at the moment I think.
Some links about capture cards:
https://obsproject.com/forum/resour...-streaming-or-recording-and-do-i-need-one.10/
http://www.helping-squad.com/wp/what-kind-of-video-splitter-should-i-get/
Last but not least, before you buy a capture card you can always ask us if it works good with OBS or has any known problems :)