With the file that you provided, we were able to find an old registry key setting that looks like it was hiding the Engine’s attempts to properly register with Windows. Click on the Windows start button, and enter regedit in the Search Programs and Files field.
Locate the HKEY_CURRENT_USER\Software\Classes\.uproject registry key and delete it. Close the Registry Editor and go pull up the Properties window for your .uproject file. Click Change to select the program to use to open .uproject files. You should see Unreal Engine as an option now (if you see two Unreal Engine options, select the one with the blue icon).