instrumentation amplifier output noise
It is well-known that Inamps can suffer from degraded performance in the presence of EMI. Here is a document from ADI that gives tips on this subject and below is a small extract: -



Of vital importance (of course) is the requirement for the engineer to read the data sheet and follow specific guidelines for the chip such as adequate power supply decoupling and layout methods. A lot of EMI issues are caused by bad layout and generally not following the data sheet.

I'm not saying the OP hasn't followed the guidelines in the data-sheet of course but it would be interesting to see a picture of the PCB layout. If it's a breadboard then basic EMI protection is out of the window!