In TimeoutHandler, use a request whose context has been configured with
the handler's timeout

Fixes #20712

Change-Id: Ie670148f85fdad46841ff29232042309e15665ae
Reviewed-on: https://go-review.googlesource.com/46412
Run-TryBot: Tom Bergan <tombergan@google.com>
Reviewed-by: Tom Bergan <tombergan@google.com>