I think you need to begin by redefining your research question. What do you mean by "inspired/influenced/affected?" These terms are too vague. For example, a friend of mine could inspire me to write a book because of something he experienced. I was so moved by the story, that I felt I needed to commemorate it somehow. Or, in another sense, a friend inspired me to write a book because he was supportive and enthusiastic about my talents.

So, in what way you looking for "royal inspiration" in Shakespeare? Are you looking for examples of when Queen Elizabeth instructed Shakespeare to write a certain play? Are you looking for examples in which Queen Elizabeth's reign motivated Shakespeare to write his plays in a certain way (ie as a critique of the Queen)? Are you looking to see how royal subventions supported Shakespeare and his company? Are you looking for royal interference and censure? Being specific about these questions will draw you closer more specific answers.

I'll give you a few leads: There is a story that Queen Elizabeth liked the character Falstaff so much that she asked Shakespeare to write a third play featuring the character. So Shakespeare wrote MERRY WIVES OF WINDSOR (this story is apocryphal, however). Another idea is that Shakespeare wrote Richard III -- which is about the start of the Tudor line -- as an ironic commemoration of Queen Elizabeth's reign. Since she was unwed, it was very clear that the Tudor line would expire with her. Then there's a third tale that Shakespeare's company was paid to put on a performance of Richard II the same night that an assassination plot against the Queen was underway (it failed). These are moments where Elizabeth symbol as monarch certainly had something to do with Shakespeare's work.

You might also consider legal ramifications. In the late 1570s, Elizabeth instituted a law that no players could perform in London without the sponsorship of a court official (thus, Shakespeare's troupe was called the Lord Chamberlain's Men -- and then switched to the King's Men after the ascension of King James). In what ways did this sponsorship privilege Shakespeare's company and allow him to succeed in London?

Check out Andrew Gurr's PLAYGOING IN SHAKESPEARE'S LONDON (http://books.google.com/books/about/Playgoing_in_Shakespeare_s_London.html?id=7CMwPTM1Ca0C) as a starting point. Read the Bibliography for places to look further. There are many, many books written on this kind of stuff, and if you haven't found any of it, its because you aren't looking in a library.