Because if she does, as some are calling on her to do, there is a zero percent chance the Senate would confirm a new HHS Secretary. This would be a more contentious appointment than director of BAFTE, which went unfilled for years. Unless Harry Reid invokes the nuclear option, HHS would remain without a Secretary.

And within Obamacare, there are a number of responsibilities that only a Secretary of HHS could perform (as I understand it). If HHS has no confirmed Secretary, then even more portions of Obamacare can’t work–in particular, the ability to grant various waivers and extensions. So yeah, there is no way she could resign. Obama is stuck with her.