Your choice of amplifier seems OK, but I wonder why you have chosen an instrumentation amplifier. An instrumentation amplifier is a bit more complex than a basic op amp and its virtues are mainly in much better balance and offset precision. For interfacing your coil, such precise balance is not necessary. If I were searching for an op amp, I would focus my attention on the noise performance specification for each one and choose the lowest noise model I can find. Here is an application note about op amp noise that you might find useful:
http://www.linear.com/pc/downloadDocument.do?navId=H0,C1,C1154,C1009,C1026,D6539
As for the coil itself, I looked around on the net for some instruction but only found these, which are not complete and they tend to discuss the issue backwards from your application. I mean that they look at magnetic coupling as a source of noise and wish to minimize, whereas in your case you want to maximize it. But the theory is still useful.
Can I suggest that experimentation is a practical way to learn how to realize the coil? You are probably limited to a certain maximum size, which puts an upper bound on the coil diameter. Find a ferrite bar or unsheilded ferrite core that might fit this size and then find some fine magnet wire (copper wire with only a varnish insulator to keep it very thin) and go ahead and wind as many turns as you can around a single axis (not in a toroid shape). Then, to see how effective it is, attach an oscilloscope to the terminals of the coil and measure the output voltage with your coil at a chosen distance from the transmitter.
Here's the links:
https://www.electro-tech-online.com/custompdfs/2008/10/MagFieldCoupling.pdf
https://www.electro-tech-online.com/custompdfs/2008/10/emc99-w.pdf
As you can see, the theory is very simple. The output voltage from your receiver coil is directly related to coil area, number of coil turns, the permeability of the core, the mutual inductance between transmitter and receiver (which is affected by distance of course and don't forget that the coil orientation is also important) and the rate of change of the current in the transmitter.
Another source of information and perhaps coils too might be those who make those time-clock receivers. These receivers operate at 60 KHz and are put inside those so-called "atomic watches". They usually use a very small ferrite bar coil. Here is a very good application note:
Technology - Antenna Design
You can search for more using keywords "radio controlled time" to find more examples and information about these low frequency magnetic receivers.
Another source of info on the coil can be found by googling using keywords like "magnetic field probe" Such probes are used to measure magnetic fields, working similarly to what you are doing, and so the theory of how they work and how to calculate output voltage would be useful.
I also wonder if perhaps a "hall effect" device might not be able to act as a receiving sensor for your magnetic field. I have not worked with these except with DC fields, so I'm not sure they are useful here.