The previous poster already linked what I'd suggest for installing IE on Linux but it is important to understand that running IE on Linux is essentially a hack. You should not assume that IE installed via this method will work as well as or exactly the same as IE on Windows.
You may find an easier solution by installing a Firefox add-on which can make Firefox pretend to be IE. https://addons.mozilla.org/en-US/firefox/addon/59
It may or may not work though depending on how the webpages in question work. Any webpage which refuses to work using Firefox but then does work just because Firefox is pretend to be IE is badly written because it's deciding whether or not your browser can handle the page based solely upon how your browser identifies itself rather than carrying out tests to see whether your browser supports the methods it requires to work. As the aforementioned add-on demonstrates, browser's can easily lie about what they are.
I'd encourage your Husband to file complaints with any Hotel which requires use of Internet Explorer in order to connect to their wireless networks. Internet Explorer is Windows only software - they are in effect locking out anyone who chooses to use an operating system other than Windows. Better still is if he tells the Hotels that he will not stay with them again as a result. (Though such a threat may of course not be practical to make good on.)