As explained in Tarkin the Jedi temple was built atop a Sith shrine.
This was slowly corrupting the power of the light side and clouding the vision of the Jedi who lived there, this would most likely make it less painful for Sidious to be there.
It is also mentioned in the book Plageius how Palpatine is able to shield himself so he doesn't show any force sensitivity. One might conclude that this talent would work the other way and shield him from any effects the light side might have on him. It would also explain why the Jedi council could not sense him