Had the landlord of that hotel heard of the disappearance of Kershaw through the usual channels, he would have put himself in communication with the police.

Sir Arthur Inglewood produced him.

How did Sir Arthur Inglewood come on his track ?" "Surely, you don't mean ?" "Point number four," he resumed imperturbably, "Mrs.Kershaw was never requested to produce a specimen of her husband's handwriting.

Why?
Because the police, clever as you say they are, never started on the right tack.

They believed William Kershaw to have been murdered; they looked for William Kershaw.
"On December the 31st, what was presumed to be the body of William Kershaw was found by two lightermen: I have shown you a photograph of the place where it was found.