Start by running the Start menu troubleshooter http://aka.ms/diag_StartMenu

Lets start with the basics:

Method 1: Run the Search and Indexing Troubleshooter.
Follow the below steps:
 1. Open Control Panel.
 2. Select Troubleshooting.
 3. Click on Search and Indexing Troubleshooter.
 4. Run the Troubleshooter.

Method 2: Run the DISM Tool. If the issue persists, I would have run the DISM tool to check the system health and will try to restore the files.  
 1. Click on 'Start'.
 2. Enter 'Command prompt' in the Search box.   
 3. In the list of results, swipe down on or right-click Command Prompt, and then tap or click 'Run as administrator'.  
 4. In the Administrator: Command Prompt Window, type the following commands. Press the Enter key after each command:
DISM.exe /Online /Cleanup-image /Scanhealth DISM.exe /Online /Cleanup-image /Restorehealth
To close the Administrator: Command Prompt window, type Exit, 
and then press Enter.
Note: The tool might take 15-20 minutes to finish running, so please do not cancel it. 

Method 3: Restart the Cortana Process in Task Manager

Press Windows key + X
Click Task Manager
Select the Processes tab

Scroll through the list of processes, select Cortana then click End Task.

Restart Windows 10 then see if it starts working.


Method 4: Reinstall Cortana\

Press Windows key + X
Click PowerShell (Admin)

At the command Prompt, type then following the hit Enter.
Exit Powershell the restart

Get-AppXPackage -Name Microsoft.Windows.Cortana | Foreach {Add-AppxPackage -DisableDevelopmentMode -Register "$($_.InstallLocation)\AppXManifest.xml"}

Source/Credit: Microsoft Forum Moderators