At long last! A breakthrough.
I bought a decent titanium drill bit this week and borrowed a friend's sledgehammer and good news!
Turns out the hammer thing actually worked. I drilled some smaller holes around the stuck drill bit to loosen it up and gave it a good whack with the sledgehammer, flew right out.
Anyway, the drill bit's out now and I've replaced it with a network cable so I won't hopefully have to worry about neighbours interfering with my wifi again.