What you are referring to as the brake shaft is actually called the motor shaft because it goes thru the motor portion of the hydro.
In the first drawing, you can see your brake cog (#52).
If you follow the line you will see that it ends up on the shaft (#33).
The other drawing shows the shaft as #73.
In the first drawing, you will see on the shaft a part #31, or part #71 in the second drawing.
That part is a snap ring, it is located on the inside of the side case of the housing.
My guess is that your snap ring has come off or is broken.
To inspect/fix would require removal of that side cover which would likely require removal of the hydro from the mower.
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