My first response for technical problems like this has always been, "how did biology solve it?"
In this case, it kind of didn't. Sleep isn't a charging cycle, as much as it is a cleaning time. Spend the less-productive night hours squirreled away, healing your body and indexing the day's memories and such. What people actually do to get their energy is eat and digest, and humans only store enough energy for about three days of actually doing things. When someone eats, they are getting all the energy from the storage of whatever they are eating. That's why you eat your own body weight in food in about a month, but it's split into a pound or two three times a day: the thing you're eating also didn't solve the energy storage problem, and you use energy almost as fast as you get it if you're an active person. It's like that all the way down to the plant level, when they use the sun's energy to turn ground-minerals into chemicals they can use.
Considering that billions of years of evolution managed to get energy storage for three days of activity, the best option probably is to have some form of on-site power generator and a steady supply of fuel, as it is much easier to generate power than store it. The reliance on fossil fuels is because they have something like 40 times the energy per kilogram of current batteries. Those biobatteries that >>797
mentioned show great potential if they can be miniaturized to fit inside of a torso.
As for artificial hearts, those work via "inductive coupling" - basically, if you have two conductors set up right, you can wirelessly transfer energy short distances, as in, a few feet at most. The transmission speed decreases very drastically with range. The heart is connected to one of the conductor (along with a rechargeable battery), while outside of the body is a much bigger battery connected to the other conductor. The rechargeable battery inside has only about half an hour of charge, almost all energy goes nonstop from the external battery to the heart. Inductive coupling can be used to transfer energy without breaking the skin, but doesn't have the flexibility to be much better than regular charging in terms of range, and takes up a lot more space, since a bigger conductor means faster power transfer at a much less efficient trade than bigger charging cables.