There are loads of ways to do this, it really depends on how neat you want things to be and whether you're prepared to spend any money to tidy things up a bit.
Were you planning on running one cable between the modem and router anyway? If that was your plan, I'd honestly just run two instead and put one back to your desktop. Do two cables really look less tidy than one? Run carefully with a few cheap cable ties I can't see why two cables would be noticeably worse than one. A single network cable is good for 300ft under most circumstances, so there's literally no impact on performance or latency whatsoever. That'd be my option.
A range extended is a completely wire free option, but I haven't had great experiences with range extenders. YMMV of course.
As @hamperking suggests, you could run a powerline adapter back to the desktop... but that's extra equipment and points of failure and the only plus is that it allows you to run one ethernet cable instead of two. I don't really get the benefit.
If you really are wedded to running one cable only, a more robust solution is probably to pick up a cheap router with a couple of LAN ports which you leave in your office and which maintains your Internet connection, DHCP/DNS etc, and acts as a switch for you ethernet-connected desktop. You can then run your single cable to the wireless unit in the more centralised location, which you set up to function as a simple access point.
Final question. What's your current wireless router? Because 15ft isn't all that far. Newer wifi standards (like Wireless AC) and higher end units with better antennas usually provide significantly better coverage than cheaper units, or even older high end units. You *might* find that just upgrading to a newer wireless router would give you the coverage you need without having to run cables or extenders at all.