How can I get Elysia Cron to run every 15 minutes?
Given the explanation in your question about how you've set things up, at least one reason your external cron does not ping Elysia Cron is that you've got the key wrong, or that the key is missing.

You need to up look the right key. Navigate to Elysia  » Settings  » Installation Settings  » Cron key and copy the key has you find there so that your Cpanel setup to looks like this:

http://example.com/sites/all/modules/elysia_cron/cron.php?cron_key=ZvMtg5HAaRzjz5NtkwoDs0F3jOWrbpIX5JVi3ACJl7Q


Note: Use the actual key you find on your site - not the dummy key I use as an example.