They did the experiment using a detector isolated by lead bricks, and that detector has a window (I forget what kind just off the top of my head) that detects specific kinds of radiation and quantifies their energies.
What I am proposing is that a more robust methodology would have been to run two experiments at the same time. You run, on the one hand, the isolated detector with the lead bricks, and the sample of Manganese 54 that you want to measure the activity of. And on the other hand, at the same time, you run an identical detector (for most purposes, a detector of the same model and year will suffice, but the closer to being identical, the better), with the same kind of window, and isolate it as well, also using lead bricks (again, the closer to being identical, the better), and run it for the same duration, and at the same time, but without the sample of manganese 54 - effectively, you're measuring the background rate. Ideally, you can do a blank run with both detectors before beginning the experiment, and prove a correlation, then subtract the background signal from the experimental signal, and see what you have left.
Finally, a smart experimentalist would also have used multiple radionuclides, and perhaps a variety of decay paths to give a truely robust result.