you can just put a pin through the holes. if you want to make it more secure, use the tip from a toothpick or the pointy end of a dental floss toothpick to keep it in place.
I managed to connect to the router over the UART connection (to USB and to my PC).
The router spits out the terminal messages. I just don't seem to be able to interact with the router from my PC using Putty. Like i cannot interrupt the boot process by pushing 'enter'.
Since you can see output, the RX pin and GND from your adapter is connected well, but since you can't send, then the TX pin (from your adapter, RX on your router) is probably not connected well.
There's lots of good suggestions on improving it, (fishing wire, plastic toothpick) perhaps find something you can wedge carefully in the hole, preferably plastic so it doesn't leave a residue and making the connection worse.
You can also try moving the cable a bit, I try to get a little tension on the cable in the right direction so that the pins gets pulled sideways and are not sitting loosely in the hole and can make a good connection that way.
If nothing else works, soldering a header is the most stable, but has risk of accidentally damaging the board, depending on how good you are at soldering.
With some practice you'll get a good connection without a header.
Using fishing line as a "tool for jamming pins tightly" sounds like a nice/fun idea. Practical tip
I might use that with some of my newer routers that have only holes.
(Any narrow plastic thread should actually work similarly.)