SQL Server Services empty in SQL Server configuration
I did the upgrade on my laptop with SQL Server 2008 R2 and SQL Server 2012 already installed. The issue I had was that SSCM actually disappeared from the Start Menu, if after repair it would not put them back. I ended up finding them under the Windows root directory, but have since done a complete reinstall of Windows 10 on my laptop, and reinstalled SQL Server.

I would expect you are looking at a similar situation that is going to require a reload of Windows 10. You might try just doing the refresh, where it does not completely remove everything...but I doubt that is going to work. I did a factory reset where I went back to Windows 8.1, then forced an upgrade to Windows 10. Which I might add, doing that seemed to have improved Windows 10 at least on my laptop (Lenovo Y 40-80) compared to running it after the upgrade.

One last chance you might attempt is to install the SSMS preview (standalone of SSMS) from here.