elemind

Plugin will allow you to fetch the WP REST API call with a “lang” variable. 
It will retrieve the right content from WPML.
Tested with the latest version of:
<ul>
<li>WPML 3.2.7</li>
<li>WP REST API 2.0-beta4</li>
</ul>

WP REST API multilanguage (over WMPL)

Allows you to request a language with your WP-API and WPML site.

digitalissimo

If you need to get the information from another site or your site using JSON API (https://wordpress.org/plugins/json-api/) you can use this plugin. 
For slideshow this plugin use Cycle2 (http://jquery.malsup.com/cycle2).
<h4>Shortcode</h4>
<pre><code>[getapi url="www.mysite.com" limit="5" type="[CYCLE or LIST]"]:
</code></pre>
<ul>
<li>
url: url before ?json=get_recent_posts
</li>
<li>
limit: number element of json result
</li>
<li>
type: “Cycle” is a slideshow of thumbs or “List” is the list of post titles
</li>
</ul>

Get Json Api

Retrieve the results of the API of a site that uses the plugin JSON API

WP REST API multilanguage (over WMPL) vs Get Json Api