Noise mitigation in +/-5mV instrumentation amplifier?
The simple answer is to buy the right IC for the job.  You need an instrumentation amplifier -- or at the very least, a specialized differential amplifier on a chip, with both the positive and negative inputs having the same gain.

Assuming your 60Hz noise is uniformly present on both inputs, this would be common mode noise.  Thus, you need a circuit with a good Common Mode Rejection Ratio (CMRR). Your circuit does not have that, by design, as you have different gains for the positive and negative inputs.  Even if the inputs had the same gain, it is still fairly difficult to construct circuits with good CMRR's because of tolerances on resistors, but it is very easy to buy them.

I suggest you confirm this for yourself in sim, by putting a 60Hz voltage source on both inputs.