Wireless Induction coil wired to USB
You could probably get some success by opening the infinity charging base and wire some extra coils to it. Afterall, it already has all the necessary electronics and I'd be surprised if those things used something as sophisticated as QI, it's probably closer to a transformer sawn in half with some basic electronics in the base to drive it at a frequency much higher than the mains (something like 20kHz vs 50/60Hz), probably not much different to the innards of your phone charger.