Yes, Brougham was in Westmorland. Only just! The castle stands on the southern bank of the River Eamont near Penrith. The northern bank and the town of Penrith were in Cumberland. From where it leaves Ullswater to its junction with the Eden the river formed the boundary between the two counties.
Brougham Castle was in the seventeenth century owned by the Clifford family and it was here that the great northern castle restorer, Lady Anne Clifford, died in 1676. In addition to Brougham, during the later years of her life she substantially rebuilt and renovated her Westmorland castles of Pendragon, Brough and Appleby.
As you can see, this second photo was taken from high up in the keep. And no, I was not trespassing when I took it. English Heritage, who care for Brougham Castle, do allow visitors to climb the narrow twisting stairway up to the top of the keep. I wouldn’t advise it for anyone suffering from anything other than the very mildest of vertigo but if you can get there the views in all directions are well worth the climb. The river seen flowing into the Eamont below the castle is the Lowther.