OP actually made me curious on his project It's not everyday you need to fit like 10 LEDs within such small surface areaI'm dubious you could fit what you want in the size of a five pence piece anyway, never mind a battery as well - a multicoloured LED (like the WS2812) is 5mm square on it's own, and a 5p is only 18mm diameter.
Just make sure that whatever you choose is indeed rechargeable. Most small lithium coin cells are not.I can't say to much but I just want to get info on the smallest controlable led and rechargeable battery available.
It would be similar to modern head phones. Individual chargeable units but with a multi coloured led to illuminate clear plastic.