Okay, I found a great solution for me. First, I think a better plugin than "syntaxhl" is called "codemagic".
https://github.com/tinymce-plugins/codemagic
After downloading, I renamed the file from 'codemagic-master' to 'codemagic'. I then placed this folder into:
framework/thirdparty/tinymce/plugins/
Next, I added the following line into mysite/_config.php
HtmlEditorConfig::get('cms')->enablePlugins('codemagic');
Lastly, I needed to add the button to the TinyMCE button layout - ensuring that 'codemagic' was listed. This too was added to the mysite/_config.php file.
HtmlEditorConfig::get('cms')->setButtonsForLine(1, 'undo', 'redo', 'bold', 'italic', 'underline', 'strikethrough', '|', 'justifyleft', 'justifycenter', 'justifyright', 'justifyfull', '|', 'styleselect', 'formatselect', 'fontselect', 'fontsizeselect', '|', 'forecolor', 'backcolor', 'styleprops');
HtmlEditorConfig::get('cms')->setButtonsForLine(2, 'codemagic', 'fullscreen', '|', 'cut', 'copy', 'paste', 'pastetext', 'pasteword', 'spellchecker', '|', 'bullist', 'numlist', '|', 'outdent', 'indent', 'blockquote', '|', 'image', 'link', 'unlink', 'anchor', '|', 'selectall', 'visualaid', '|', 'removeformat', 'hr', 'charmap');
I really hope this helps the others who were looking for an answer to this problem.