Updated latency fixture to match internal classes and refactored
connection helper.

1. Moved the Open new document logic to the SetUp function to enforce
a fresh document for tests.

2. Refactored connection helper to make it as minimum as possible, and
moved the ping response verify logic to the connected client fixture.

BUG=

Review URL: https://codereview.chromium.org/1253613002

Cr-Commit-Position: refs/heads/master@{#340588}
14 files changed