So... not an entirely successful day. Being the absolute idiot that I am, I managed to mistake 4.7k resistors for 47k ones. My power supply was working nicely giving me both a positive and negative output, then I noticed a few things (in this order):

1. The negative voltage was about 0.5V too large.

2. The negative supply suddenly became quite noisy.

3. There was a burning smell.

4. The negative supply was no longer negative but positive.

Conclusion: The circuit will almost certainly work if I use the correct resistors.

Future work: Get a new LT1054.