I'm having what appears to be the same problem. Note that this problem only occurs when I use Internet Explorer on the desktop. The Metro IE works fine.
However, a unique twist. I have a Windows 8.1 Desktop, Laptop and Tablet, all updated. I cannot search using Bing within IE11 on either my Tablet or Laptop. Nothing happens when I try to type in the Bing search area. It doesn't matter whether I'm using the Touch Keyboard or the real keyboard on my laptop. I can click on favorites and go somewhere else and everything works, including other search engines. Also, when I'm on bing.com, I notice that all of the various options in the Gear Icon are greyed out except File, Safety, and Compatibility View Settings. I can type in the address bar and that works fine. If I go to another web site, all of the Gear Icon options become available.
This doesn't make sense to me. Any thoughts.
By the way, when on another web site, I went to Internet Options and deleted everything but passwords and tracking info. Did the same using CCleaner. Also, if I go to bing.com using Firefox as my browser, the Bing search works fine. The only problem is when I try to use bing.com search on my desktop using the desktop ie11 browser.