Middleware example from Railscast #151 doesn't work as expected. Need help.

My middleware class "ResponseTimer" doesn't intercept a request. I don't
know why (the example is from here
http://railscasts.com/episodes/151-rack-middleware?view=asciicast; I
changed my one a little for Rails 3.2.13 version, the original is for
Rails
2.x).

So I did as follows:

## in \lib\response_timer.rb
## this is the class of my custom middleware:

class ResponseTimer
  def initialize(app)
    @app = app
  end

  def call(env)
    [200, {"Content-Type" => "text/html"}, "Hello, World!"]
  end
end

## in \config\initializers\custom_middleware.rb
## this is an initializer file that I created. It's goal is to add my
## middleware "ResponseTimer" to the middleware stack:

module Demo
  class Application < Rails::Application
    config.after_initialize do
      Rails.application.config.middleware.insert_after(ActionDispatch::Callbacks,
"ResponseTimer")
    end
  end
end

Now I check whether "ResponseTimer" is in the middleware stack, and it
is:
C:\myapp>rake middleware
...
use ActionDispatch::Callbacks
use ResponseTimer ## ok, it is in the middleware stack
use ActiveRecord::ConnectionAdapters::ConnectionManagement
...

But when I make a request my "ResponseTimer" doesn't intercept it. The
request successfully passes by to the application. Instead of
ResponseTimer's "Hello World!" response I get standard response:

Hello#index
Find me in app/views/hello/index.html.erb

What do I do wrong? What did I forget to set or enable?