Like innumerable other Win10 users, I have been trying for quite a while to get broken apps and Cortana to work again, after the Anniversary 'update'. I've repeatedly tried sfc /scannow; DISM restorehealth; MS Repair tools; Windows troubleshooter; the infamous Powershell "Get-AppXPackage | Foreach {Add-AppxPackage -DisableDevelopmentMode -Register "$($_.InstallLocation)\AppXManifest.xml", ad nauseum, with no luck at all.
Today, while Googling a somewhat different topic, I stumbled across a shareware program that actually FIXED Cortana, and ALL metro apps. It's called Windows Repair Free/Pro, from Tweaking.com. The free version is fully functional for repairing Windows problems; just follow all the directions exactly. The $20 Pro version just adds some performance and visual tweak options.
You have to run the repair tools twice; it took about 4-1/2 hours, total, on my system. On the first pass, when the repair tool was re-installing metro apps, several of the same errors appeared in the cmd box as with the above mentioned Powershell command line. After the required reboot into Safe Mode w/networking, the second pass returned no errors at all, and everything is working again.
If only I had found this program five or six weeks ago! I could have saved countless frustrating hours of beating my head against the screen.
If Microscrew had programmers with this level of talent, they might be able to produce a reliable product.
BTW, I was so impressed with the free version, I bought the Pro version, just to support these guys, something I very rarely do.