I have the following packages and their dependencies installed in my WebAPI project:
Ninject.Web.WebApi
Ninject.Web.WebApi.OwinHost
I too was trying to make an 'only API' app, but faced similar issues. For me personally, in the end it came down the selecting the right starting template.
Despite going for API-only, when in the 'Create a new ASP.NET Web Application" wizard, selecting the 'Empty' application-type and choosing MVC and Web API in the 'add folders & core references' section made Ninject work fine:
As opposed to the 'Web API' application, which didn't play nice with Ninject:
After that, it's simply to add the Ninject.Web.WebApi.WebHost package, which adds the NinjectWebCommon.cs class, and start adding dependencies.