To do the HDR merge in Lightroom you have to have Lightroom CC or Lightroom 6. That feature is not part of Lightroom 5. Never has been, and never will be. It looks like you are using Windows 10. If that is the case then you and went to Settings/System and then click on About at the bottom of the list and it will tell you whether you have a 32-bit or 64-bit operating system.
I'm wondering, however, if maybe you have installed Lightroom CC and you previously were using Lightroom 5. The two programs can be installed on the same computer and they have the same desktop icon. Is it possible that you have installed Lightroom CC, but are still using your Lightroom 5? Since your "Edit in" menu gives you the option of editing in Photoshop CC 2017, I think you have installed Lightroom CC 2015, but the desktop icon you are using is still pointing to your old Lightroom 5. What happens if you open the creative cloud application manager? Do you have the option to run Lightroom CC?
Lightroom 5.7.1 was the last update of Lightroom 5. Merge to HDR was introduced in Lightroom 6. If you are using Lightroom CC and are still on 5.7.1 then you either haven't installed the upgrade or you have a 32-bit operating system. Lightroom CC/6 is 64-bit only.