Restore detect_idle_timeout client setup and cleanup

[ac1982]

Fixes #119.

detect_idle_timeout still passes a positional client name even though initialize accepts keyword arguments, and its ensure references a variable defined only inside the connection block. The resulting NameError prevents the helper from returning an idle interval.

Initialize the counter in method scope and pass name: to the client constructor. Keep the persistent client separate from its underlying Net::HTTP connection so its pool can be shut down on completion or a handled error.

The four regression tests run the real constructor with a substituted connection and sleep implementation: reaching the maximum, a zero maximum, EOF after a successful request, and a non-200 response. They verify the returned interval, sleep sequence, and client shutdown. All four raise NameError before the fix and pass afterward.

Validation (Ruby 3.3.8, Linux arm64):

• Focused tests: 4 runs, 4 assertions, no failures/errors.
• bundle exec rake test manifest:check: 119 runs, 322 assertions, no failures/errors, 1 skip; manifest check passed.
• git diff --check passed.

No live-server idle-timeout measurement was performed; the existing warning about modern Ruby behavior remains. Other Ruby versions and optional net-http-pipeline support were not tested locally. AI assistance: OpenAI Codex investigated, implemented, and ran these checks.