My first hit was a forum post from June 2014, some of the advice is out-of-date: https://productforums.google.com/forum/#!topic/business/Cdxw...

This was one of the most helpful posts:

  Google has a bias against web designers and seo in the scheme of things. So things that are easy for your clients are not going to be as easy for you. 

  There are some things that will increase the likelihood of the panel showing (although these are no guarantee by any means)

  1- Use Schema formatting for your address information on your site

  2- Work to increase your businesses location prominence by getting recognized across the internet, get more citations ( you have very few), links and reviews

  3- Build up followers on G+. 100 is a number that I have heard bandied about but quality of followers trumps quantity. 

  As an aside Google prefers location based businesses to NOT use a profile photo for their business. They prefer a shot of your store front or perhaps an interior shot. 

  Also be sure that you are in compliance with Google's naming rules offline and on. You have some keywords in your name which might or might not be in violation depending on how you are legally named, how you are known in the market place, how you answer the phone etc

So it looks like an algorithmic thing, not something you would sign up for.

Also, this help page was suggested, it has a lot of suggestions: https://support.google.com/business/answer/3038177?hl=en

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Edit: This forum post suggests making sure Wikipedia is up-to-date, as that's one of the Knowledge Panel's sources: https://productforums.google.com/forum/#!topic/business/JsEL...