Switch to using Bundler for building gems

All philosophy aside, Jeweler has issues in regards to newer versions of
RubyGems and Bundler. One is the dreaded

    $ rake build
    WARNING:  licenses is empty
    rake aborted!
    private method `open' called for Gem::Package:Class

Others are reported and documented at https://github.com/technicalpickles/jeweler/pulls.
Bump JSON dependency

Without this we'd get a warning on every invocation of mite:

    $ mite
    You are using an old or stdlib version of json gem
    Please upgrade to the recent version by adding this to your Gemfile:

      gem 'json', '~> 1.7.7'
Bump RSpec to newer version
Get rid of RSpec 2 deprecation warnings
Clean up formatting
Don't use the users actual mite.yml configuration when running tests
Don't send requests to the API endpoint in tests
Make Rspec rake tasks work again