Good luck / my condolences.
Weird. I can't say that I've EVER seen SPEAKER WIRE that's SOLID
and even slightly resembles the thickness or stiffness of CPU pins.
But you're 100% right, you want something that's as THICK as the CPU pins
to very close tolerances, maybe slightly thinner if you have to error on one
side or the other, and with high stiffness.
Gold does NOT matter. I'd look for hard drawn copper or even steel wire if you
can find it in the right thickness. Sure it'd be nice if it wasn't subject to
corrosion and had high conductivity like gold or copper, but those are softer
metals and it's useless how pretty & corrosion resistant it is if it doesn't
work at all or reliably.
If you're going to stick it in the socket hole and HOPE it makes contact you'll need
to cut the length PRECISELY right to be a tiny bit too long but not more than
15% or so beyond the normal pin length so it cannot short out to the next pins over.
It'd be preferable to solder the wire onto the CPU itself but you'd need a good
magnifier lamp, fine point low power soldering iron, good ESD control, etc. to stand
a good chance of that. But if the broken pin is on the edge or near a gap you may
be in luck to try.
Look in the CPU electrical data sheet for the FUNCTION of the pin.
If you're SUPER LUCKY it's a "No Connect" pin that is useless.
If you're SUPER LUCKY even if it's not a "No Connect" it'll be one that isn't needed
in practice like one for factory testing or so.
If you're PRETTY lucky it'll be a power pin with the same voltage and function as
one that's right next to it in which case you can just use solder or conductive
trace repair ink to short it to the "same function" pin next to it and that's enough.
If you're a gambling man and get lucky it may be a power pin that may be 'alone'
in its immediate area but if there are several other power pins that power the same
function in the chip elsewhere in the package, maybe they're all connected together
inside the chip and it'll work out with one missing.
If you're in the mood for humor, hey, just leave it broken and never run 64
bit software, you've still got 63 good bits!

j/k
Actually it's possible it does go to a RAM stick or RAM pin that you don't actually use
in your system depending on what the pin does and what broke.
Usually the ethernet cable I see that's intended for wiring inside the walls
is solid core around 22 or 26 ga. wire so actually that could be stiff and work
if the thickness is similar. Maybe the flexible "patch cables" tend to use
stranded (I've never seen that though) or flimsy / too thin wire though.
Paperclips, Radio Shack, or MAGNET WIRE that you scrape the insulating varnish
off of should work.
Good luck!