This start button issue happened to me too. Don't know why, as it continues to work fine on other machines. However, for this particular PC I couldn't update Windows 10 either to 1511. It just kept on failing, so obviously there was something borked about it.
Eventually I just "upgraded" my Windows 10 which was an older build to Windows 10 1511 using the 1511 DVD install disk. That worked, and fixed the start button issue too.
Even though it was a Windows 10 to Windows 10 upgrade, it left me with a Windows.old directory just like if I had upgraded from Windows 7. The benefit is it kept all my applications and setting intact, but with a fresh install. I wonder if this is a better solution than a Windows 10 repair.