I just ran into this same issue. I however could login to the web client with a domain account, and administrator@vsphere.local. This was after an upgrade. I can't confirm whether or not patches were applied prior to upgrading. BWillburn, what Microsoft patches did you apply? Maybe if we compare we can figure out which patch broke vCenter.
I systematically upgraded every component, and verified after each upgrade that the corresponding component did not break my vSphere environment. SSO, Web Client, and Inventory Service had no hiccups, logged in to the web client and the c# client with no issues. Once I "successfully" upgrade the vCenter component, that is when hell broke loose. My Recent Tasks was throwing "Unable to create the managed object for - urn:vmomi:TaskManager:TaskManager:1086A58C-13EA-4A65-BF3D-3A468AB1CB59. Also was getting the "Could not connect to one or more vCenter Server systems: https://vcentername.com:443/SDK"
I uninstalled vCenter, leaving existing certs and SQL database intact, then re installed. Once I re installed, the issue was resolved. I don't know why this happened, if I had to guess the upgrade checker missed some "bad data" somewhere with the vCenter component which left vCenter in this unhealthy state.