We were able to find a partial solution to this issue. By adding the following code to both header.jsp and header-deprecated.jsp files in the includes\decorators folder we are able to enable the correct Document Mode.
<meta http-equiv="X-UA-Compatible" content="IE=edge" />
However, that code needs to be the first thing after the <head> tag in both those files in order to take effect. Apparently the 'edge' value is not recommended for production use, but it seems to fix a lot of the issues.
However, it does cause a new issue when using IE8. This is because IE is still defaulting to a browser mode of Compatibility View. The issue that is still seen is when searching for issues, the issue list has columns which are extremely wide so therefore very difficult to find the correct issue.
I think this all has to do with the way Microsoft controls it's compatibility view settings. According to a non-microsoft site at http://hsivonen.iki.fi/doctype/, in the section named "IE8 and IE9 Complications" it says that IE uses doctype sniffing like other browsers only if the site is NOT in the Intranet zone. However, when accessing any site via a short name (netbios), such as http://jira, IE places that site in the Intranet Zone, and therefore enables Compatibility View which therefore changes the way IE displays the page.
So far, I have not found a solution to that besides convincing the domain administrators to implement a group policy change which will turn off compatibility view for all intranet sites. However, this could cause issues with other sites which may be legacy sites on our network, so they are very hesitant to make such a change.
Unless Atlassian has any pull with Microsoft to have them change the way they handle their Compatibility View mode in IE, the group policy change seems to be the only solution.
Also, turning off compatibility view for Intranet zone sites with group policy resolves all issues and the extra meta tag isn't even needed since IE then follows HTML standards and uses the doctype html tag to tell it to use HTML5 encoding.