Why is my instrumentation amplifier's output voltage completely wrong?
The offset voltage of each of the op-amps you're using can be as much as +/-3mV at room temperature. So, the difference between two outputs could be as much as 6mV different from the inputs with unity gain. You're seeing 5.4mV which is large, but within specifications and therefore plausible. 

Since you don't have much gain in the first stage (only 3) you also have to consider the offset voltage in the second stage. In any case, 638 times your measured differential input offset voltage of 5.4mV + 2.5mV signal is almost 5V.  

You can either use better op-amps (such as autozero or 'zero drift' types) or null out the offset voltage by some means (trimpot or reduce the gain and do it digitally). You should also consider the drift of the op-amps you're using which is not guaranteed, but is fairly reasonable typically (+/-1.7uV/K).