I am posting as an answer here from @magicandre1981 comments above, so I can close the question.
Windows SDK 10.0.26624 is the SDK for the first Build 10240 from Summer 2015.
Windows SDK 10.1.10586.15 is the SDK for the November 2015 Update 1511.
I ended up uninstalling both versions and letting Visual Studio install the necessary package to run my project, which ended up being version 10.0.26624.