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This story is no doubt related to the 1910 publication of Chesterton's book What's Wrong With the World, the dedication to which contains a number of jokes at Chesterton's expense:
Many a mild lady visitor opened her eyes when I remarked casually, "I have been doing 'What is Wrong' all this morning." And one minister of religion moved quite sharply in his chair when I told him (as he understood it) that I had to run upstairs and do what was wrong, but should be down again in a minute.
It is not entirely certain whether Chesterton did write this letter, although Dale Ahlquist, president of the American Chesterton Society, believes it's likely true. There appears to be no evidence of the letter.
This story has been repeated so often about Chesterton that we suspect it is true. Also, it seems it is never told about any one other than Chesterton. What we have not found, however, is any documentary evidence for it. It may indeed be from The Times, as the story is usually told, but no one has taken the trouble to go through the back issues and find a copy of the actual letter. It has also been attributed to other papers, but again, no proof.
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I spent the last hour scouring the last pages, of the last eight years of Gilbert! Magazine where Chesterton "answers his mail" because they always put real responses to made up questions in there, but this one was "What's wrong with the world" and his response was "The devil" and that was dated 1908.
That counters what you, popular mythos and The Guardian report:
This year, however, sees the centenary of one of his rather less high-profile publications. What's Wrong with the World represents an extrapolation of Chesterton's original response to a query posed in so many words by the Times to a selection of eminent writers and thinkers of the day. "Dear Sirs," ran GK's succinct rejoinder, "I am". The publication of the book suggested that, on reflection, there might have been more to say on the subject.
https://www.theguardian.com/books/booksblog/2010/jan/08/gk-chesterton-world-whats-wrong
Another half-heard truth possibly misremembered, but certainly worth mentioning, was that the letter you mentioned was sent to a newspaper in Tusla Oklahoma, of all places. Digging up that reference would most certainly be more difficult than me combing back issues of Gilbert!
Chesterton did on several occasions (but without repeating himself) come up with essays on the same topics as his books before writing the books. He also did, when asked a question or prompted "gonand write a book about it" so it's kind of funny that he didn't write. I think I shall write a book about it.
On the other hand, I find it a tad odd that he wouldn't have mentioned the letter in his book "what's wrong with the world".
Lastly, I have been part of an effort to digitize everything ever written by G.K.C. and it should hopefully be available in the next few years, so maybe the mystery will be solved. I did transcribe a number of mailbag-type responses from him, so with any luck this quote in question will be part of the Chesterton Digital Library, and I'll update this post if I find it, but don't hold your breath!