Inductive Coil - detecting open coil
Sensing the current thru the primary is the obvious way to detect whether it is present.  All you need is a small current sense resistor.  It looks like this could be on the low side easily enough.  Amplify that and look at it during the peak of the 2 kHz drive signal.  If it's not high enough, then assume the primary is open.

For the secondary, maybe your receiving circuit can pass a little DC current thru the coil.  That should be irrelevant to its operation, but would allow you to sense its presence, whether it is receiving any AC signal or not.  Something as simple as a high-resistance pullup on one end with the other end grounded would do.  If the DC voltage ever went high on the pulled-up end, you know the coil is open.