If it is a valuable piece (time invested, $, etc.), you can buy a tap extractor from places like MSC, McMaster Carr, etc. A good local hardware might even have them.
They have fingers that slip between the taps flutes, allowing you to remove it. Since it is aluminum, it may be fairly easy to break free, so you might even be able to make something similar and improvise (thin needle nose or round nose pliers?).
On very valuable pieces, you can try to find a machine shop that has a "tap buster" . This is basically what led to more sophisticated EDM (Electric Discharge Machining) machines that are now widely used in complex mold making and aerospace applications. This proces is not detered by a metals hardness or toughness as most traditional machining methods are. I used to use a few variatons of the EDM process in the manufacture of aircraft and land based turbine blade and vanes, i.e. Sinker EDM for creating a complex female shape into metal, wire EDM for cutting complex shapes similar to a scroll saw, and a specialized sinker/ multi wire comb method we used to create rows of cooling holes in blades.
Basically, the Tap Buster or EDM is a controlled spark erosion prosess where it "eats" the tap or other target base metal away. Much like when you strike an arc with a torch, you will notice pit in the base metal, same basic idea but highly controlled.
Anyway, mostly likely the top suggestion is the most appropriate, but I thought I'd share some other methods as well just in case or just for some general knowledge of avaialble processes if nothing else.
Drew