This time I attempted to override this file:

/components/com_k2/k2.php

I put this file in

/mytemplate/html/com_k2/k2.php

and it's not working. Why?

That is not a template override. You are trying to override the core of k2.
You can do a template and a module override with the method in the documentation above, but not the core.
Many times you can make changes to the files in your template override to do custom things, but its sometimes trickey.
What are you trying to change?