fix: Build API TypeScript for production

- Change build script to compile API TypeScript to api/dist/
- Update start script to use compiled JS

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

package.json

@@ -7,8 +7,8 @@
     "dev": "concurrently \"vite\" \"tsx watch api/server.ts\"",
     "dev:api": "tsx watch api/server.ts",
     "dev:ui": "vite",
-    "build": "tsc -b && vite build",
-    "start": "node api/server.js",
+    "build": "tsc -b && vite build && tsc -p api/tsconfig.json",
+    "start": "node api/dist/server.js",
     "lint": "eslint .",
     "preview": "vite preview"
   },