Unfortunately, there isn't a way to have dynamic json in a visualization on a dashboard. What is the nature of the json that you want to display?
If the data doesn't change very often, and you have the means to generate the document, you could always pragmatically update a markdown visualization with updated json data. In the .kibana index, visualizations are stored as elasticsearch documents. You could query and inject the appropriate json object into the document. The dashboard would always have the most recent version of that visualization.
If the data is refreshed more frequently, or don't have a mechinism to generate the visualization document programatically, you could use Kibana's plugin system. You could create a new visualization plugin.
Unfortunately, the plugin stuff isn't really well documented yet. We have a yeoman generator to get you started though:
And this little example might help as well: