fix: TypeScript errors — declare authenticate, fix WebSocket types

backend/package.json

@@ -13,17 +13,16 @@
     "@fastify/multipart": "^8.3.0",
     "@fastify/static": "^7.0.4",
     "@fastify/websocket": "^8.3.1",
-    "bcryptjs": "^2.4.3",
-    "fastify": "^4.28.1",
-    "pg": "^8.12.0",
-    "web-push": "^3.6.7"
-  },
-  "devDependencies": {
     "@types/bcryptjs": "^2.4.6",
     "@types/node": "^20.14.0",
     "@types/pg": "^8.11.6",
     "@types/web-push": "^3.6.4",
+    "@types/ws": "^8.18.1",
+    "bcryptjs": "^2.4.3",
+    "fastify": "^4.28.1",
+    "pg": "^8.12.0",
     "tsx": "^4.15.7",
-    "typescript": "^5.4.5"
+    "typescript": "^5.4.5",
+    "web-push": "^3.6.7"
   }
 }