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An answer to a similar question discusses limitations of House Elves. If House Elves couldn't be used to search Chamber of Secrets, why didn't he simply asked ghosts?
Ghosts can certainly move around freely without any bound.
If Moaning Myrtle was shy to be interacted, he could order house ghosts like Nearly Headless Nick or Bloody Baron. Also, Professor Binns could have happily accepted his request.
Why didn't Dumbledore ask ghosts to search for the Chamber of Secrets?