Ship `.d.ts` types and generate an API reference with TypeDoc

[PAMulligan]

Summary

Guarantee that consumers of the npm package get full TypeScript IntelliSense, and auto-generate a hosted API reference from JSDoc/TSDoc comments.

Motivation

Great types are the cheapest form of documentation. A hosted API reference also serves as the authoritative source for prop tables in the docs site.

Acceptance Criteria

• Vite build emits a single index.d.ts bundle (via vite-plugin-dts or tsc --emitDeclarationOnly)
• All public exports (ChatWidget, props, hooks, types) have TSDoc comments
• pnpm docs:api runs TypeDoc and outputs HTML to dist/api
• Generated reference published to Cloudflare Pages and linked from the docs site
• CI fails if any public export lacks a TSDoc summary