Yeah I kept trying. This seems impossible. I've messed around with page.php where the icons for social sharing, editing, and such are for pages. I've looked at the news_template.php for the theme I'm using and the examples you gave. I see nothing to change or add as the shortcode for the icon exists, it simply will not show up on the page, despite being able to show up on news articles. So I can only theorize that there is something stopping a print icon for a print friendly book page to show up.
I even changed the page.php? part of the web address to print.php? which is what the print layout seems to do. However nothing in the page will show up in that layout as it seems to expect a news article.
In the meantime, I've added on a firefox extension to grab a print friendly version of the pages for/if anyone needs me to print one for them. It would be nice though IF people could print them from home straight from the website. But at least until this gets figured out I have a work-around solution.