So as a solution I ordered a QI wireless charging set. A Note II receiver that claims to have charging current of 500-1000mA at 5V, works up to 5mm from the reciver at the frequency of 100-200kHz with efficiency of more than 70%.
I also bought 2 transmitters. One nice round plastic disk and the other one a bare PCB with a coil.
They both take an input of 1500mA at 5V and claim a 75% conversion efficiency.
And the other one:
It's stranded to increase the surface area to transmit more power.
Unfortunately the receiver's coil is just a copper track on a flex PCB. And has some surface area but not as much.
The charging rate is just average. Gets 460mA through. Not as claimed but good enough, falls in the 70% of 500mA. (The phone is claimed to charge at up to 2A!)