Hello James,
Soldering lead-less packages can be tricky. Have you tried washing the board after soldering? There are several components that are high resistance value. Make sure there are no evident assembly defects and then start debugging with a voltmeter. But when you do, you're going to want to use a meter in which you can disable the 10M input impedance (this is very often fixed in handheld meters but not in benchtop boxes). The reason for this is that the input impedance of the meter can load down the nodes you want to check.
For example, have you checked the REFIN input? SDN? It is very important to make sure the part is not in shutdown. Then, take a look at the REFOUT terminal to see if it has the same voltage as REFIN. Also, you should check the output of the instrumentation amplifier at the IAOUT pin, where you should observe the input signal amplified by a factor of 100. If you do see a signal here, then it is possible that the problem is with the opamp circuit. I would remove components to make it a simple follower and add them back.
Try a few of these things and let me know if you get stuck.
Good luck!
Gustavo