Just cogitating on the Q, according to theory your iron nails will reduce the unloaded Q. To increase the Q, you could try wave winding or even better "basket weaving". Then with a very high unloaded Q, the power consumed will be minimal with no load. So with a 50:1 standby to charging power and lower losses, this would improve the circuit.
One other thing I have seen to increase the directivity of ferrite aerials is to put them into a tube with a slit down the length. Why?, your coil is a short length of a solenoid and I guess that you would want to concentrate the flux within the coil, flux outside it is power wasted. So a thin strip of Ali on edge (ends not touching) outside the coil might just have the right effect, though it will reduce the inductance. I wonder if the coil was wound "flatwards", i.e. one turn, then next not above it but inside it, then another. . . So its a flat spiral, should give equal flux over its total face? etch the whole thing on a piece of PCB? Its a lot of copper to remove though!