IE11 Developer Tools does not work

As Andy Sterland mentioned in the comment, this is due to the December patch not being applied correctly. Today I installed an update in Windows again, named Cumulative Security Update for Internet Explorer 11 for Windows 7 for x64-based Systems (KB3008923), and this update seems the one that resolves the issue.

Worked for me at last!!!

Make sure you download a proper file. You need a patch for 64 bit system, not 32 bit. Microsoft site does not detect your system and allows you to download wrong file!

This is 32 bit: https://www.microsoft.com/en-us/download/confirmation.aspx?id=45134

This is 64 bit: https://www.microsoft.com/en-us/download/confirmation.aspx?id=45154