Hi,
Please make sure your ASP.NET project is ASP.NET website instead of ASP.NET web application, because App_Code folder is only for ASP.NET website project.
You can create a normal folder and then rename it App_Code and then add DataSet in it.
If we create a simple ASP.NET website and add Typed DataSet, does it have the same issue? If yes, it seems that your Visual Studio is corruption in this aspect. You can try to repair your VS.
I look forward to receiving your test results.