You can add Swashbuckle to your ASP.NET Core webapp and then generate Swagger/OpenAPI spec.
With the Swagger/OpenAPI spec, you can then use Swagger Codegen to generate not only TypeScript API clients (Node, Angular, Angular2, Fetch), but also API clients in C#, Java, JS, Swift, etc as well as API documentation.
The easiest way to try Swagger Codegen is to use https://editor.swagger.io by importing your spec and then select the target API client/server you want to generate. (https://editor.swagger.io leverages https://generator.swagger.io to generate code and generator.swagger.io is powered by Swagger Codegen)
Swagger Codegen can also generate server stubs in C# Nancyfx or ASP.NET core so you may consider following the contract-first approach by writing the spec using https://editor.swagger.io and then generating both API clients and server stubs.
UPDATE: On May 2018, about 50 top contributors and template creators of Swagger Codegen decided to fork Swagger Codegen to maintain a community-driven version called OpenAPI Generator. Please refer to the Q&A for more information.