Above is the battery after running for 24 hours. It held between a quarter and a third of an amp for at least a couple hours, and today 24 hours later it is still reading a few milliamps. These readings were taken by simply placing the electrodes of the voltmeter in each beaker. The jar in the left held a green liquid aluminum chloride and the one on the right held a whitish aluminum hydroxide. These were prepared by dissolving an excess of aluminum foil in HCl and NaOH and decanting each liquid part into a beaker. Then a tube full of water was carefully inserted between them to form a bridge. You might be able to see in the picture above that eventually an unknown to me orange precipitate forms in the one on the right and a dark precipitate forms in the one on the left.
The trick was finding an acid and a base that conduct electricity well, I found it hard to find information on the internet, but aluminum chloride has about 750 ohms of resistance and aluminum hydroxide has 14,000. So an obvious way to improve the battery would be to find a better base for conducting electricity than aluminum hydroxide.