Windows 10 suffers a major update issue. You can can check this by trying to open the Windows Store and install ANYTHING, if it fails to to install this is likely the reason Lightroom won't open -- your windows update is buggered up!
SOLUTION: Update Windows
But it isn't so straight forward, because the update service itself is screwed up. You can verify this by going to Start -> Windows Update -> Check status -> Probably some error code
This is a totally safe install, note it comes from Microsoft.com. This will take like 2 hours, reboot your computer a bunch of times. Once this is done, probably a good idea to do the regular windows updates since you're already there doing it.
THIS WILL SOLVE ALL YOUR WINDOWS 10 MICROSOFT STORE NOT INSTALLING APPS & ADOBE LIGHT ROOM NOT OPENING IN WINDOWS 10.
Source: I just did this on my desktop & laptop both suffering from the same things, now 100% working. Reply here if you STILL don't have a solution after doing this. Otherwise I hope I could be of assistance!