potassium sulfate is not alum, it is missing aluminum in it, potassium aluminum sulfate, ammonium aluminum sulfate etc. is the required chemical
alum probably oxidizes aluminum part, but can't break down the formed oxide further and there fore cannot continue to react with more pure aluminum (same defensive mechanic protects aluminum to an extent from most other offenders), iron oxides are not as inert, hence the alum being able to do its thing, that being said, if I had to do this on an aluminum part, I'd first do a test on a scrap piece, see how if any effect the heated up alum has on the aluminum, and I might also rig up some sort of container around the broken steel tool on the part, so as not to have to submerge the whole part into the alum solution, you can also apply the necessary heat for the reaction to the part this way, which, depending on the shape of the part might actually be easier than heating alum solution and dipping the part
high concentration nitric acid does work, but it is dangerous to handle and also has some effect on the aluminum, I've successfully dissolved some stainless M2 screws broken off in aluminum part and anodized the part after, some surface prep was needed after the acid prior to anodizing (the surface had some texture to it), AFAIR I used scotchbrite and non-ferrous polish, it anodized to an acceptable finish
