Re: hasn't really thought this through
I did some research and its possible to use a chip antenna similar to the ones used for Bluetooth.
In fact the ones from old piezo backlight inverters with a bit of minor modification should work as they are already resonant at around 1-3 MHz in the X direction but obviously in the Z direction it would just be a matter of connecting the edges together as Gnd and HV out as Ant In.
(scuttles off to test this with the one he found in an old Sony Vaio)