Jetpack uses your site's XML-RPC file to communicate with WordPress.com. I checked the XML-RPC files on both of your sites, and something seems to be breaking the response sent by the files when you try to access them.
The responses are in fact truncated due to the presence of a UTF-8 BOM. Could you contact your hosting provider, and ask them to have a look at the issue?
They can reproduce the problem by trying the following query:
curl -is -H 'Content-Type: text/xml' --data '<?xml version="1.0"?><methodCall><methodName>demo.sayHello</methodName><params></params></methodCall>' 'http://gestion-du-temps.jeux-fr.eu/xmlrpc.php' && echo
They will see that the response misses some characters, because of the BOM: