Interesting, David. Never knew that. With respect to the pool; I see that Priory Brook ran in an easterly direction not far North of Colmore Row and possibly was leated to there from further upstream. It would have run from the Edgebaston area. Quite a long run for a leat but not unusual in the Birmingham area and the downhill, north of Colmore, would have given a decent head for a wheel if the water volume was worth it. The last point may be why no reference can be found about a mill here and since the pool was also a private entity belonging to a forcibly dissolved order; then perhaps it could easily have been forgotten.
If the engraving and annotation of the grindstone in the file making plant is correct though, perhaps this leat was used again for just this one application which would only have needed a relatively small supply of water possibly.