It's good to see optimistic replies!

Assuming pimpernell is right, and the reason the bolt broke was that it was a re-used and therefore weakened stretch bolt, then it likely broke off under less torque than I had originally assumed. That means it is less likely to have done damage, and hopefully it will be easier to get out.

I'm also glad to hear that vise grips might do the trick. I was sceptical about this, and was waiting to hear some advice before I tried anything. But I guess I'll give that a go first and see if it works.

Four new bolts are in the mail (I'm going to replace the ones that went back in as well in accordance with pimpernell's observation), and I'll see how things go on the weekend.