Just putting the part in a solution of Alum and water isn't enough. It may take weeks to dissolve the broken screw.
I did a job very similar to yours a couple of years ago. The solution should be heated to just below simmering. You should dissolve as much alum as possible; keep adding alum until no more will dissolve. You will have to be around to make sure it doesn't dry out. If you have to leave it alone, turn the heat off.
It may take 2-3 days but it will eventually completely remove the broken screw. That doesn't mean you have to nurse it along on a hotplate for 72 hours it just means that that is how long it took mine to dissolve over a weekend.
The alum method is, by far, preferable to the acid methods. Alum is a food additive and is pretty safe to work with. No more precautions need be taken than for boiling an egg. It will not change the part dimensionally. I have no idea what it will do to the anodizing as I haven't had to use it on any anodized parts yet.
I did mine in a large Pyrex beaker. I would recommend something similar or an aluminum pot. For somewhat obvious reasons do not use an iron pot. I don't know that it would affect the efficacy but my guess would be to not use stainless steel either.