feat: add admin UI control for chat auto-archive days (#24704)

Relates to #24642 

Adds admin UI controls for managing chat auto-archive (days) under
"Lifecycle".
Also adds a "Days" label to the right of the pre-existing unitless
numeric input for consistency.

Exemplary screenshot below. More screens available in Storybook.

<img width="847" height="585" alt="Screenshot 2026-04-24 at 16 48 59"
src="https://github.com/user-attachments/assets/d38de5f8-d379-4b06-b175-ac399f31e578"
/>

site/src/api/api.ts

@@ -3379,6 +3379,24 @@ class ExperimentalApiMethods {
 		await this.axios.put("/api/experimental/chats/config/retention-days", req);
 	};
 
+	getChatAutoArchiveDays =
+		async (): Promise<TypesGen.ChatAutoArchiveDaysResponse> => {
+			const response =
+				await this.axios.get<TypesGen.ChatAutoArchiveDaysResponse>(
+					"/api/experimental/chats/config/auto-archive-days",
+				);
+			return response.data;
+		};
+
+	updateChatAutoArchiveDays = async (
+		req: TypesGen.UpdateChatAutoArchiveDaysRequest,
+	): Promise<void> => {
+		await this.axios.put(
+			"/api/experimental/chats/config/auto-archive-days",
+			req,
+		);
+	};
+
 	updateChatTemplateAllowlist = async (
 		req: TypesGen.ChatTemplateAllowlist,
 	): Promise<void> => {

site/src/api/queries/chats.ts

@@ -1351,6 +1351,22 @@ export const updateChatRetentionDays = (queryClient: QueryClient) => ({
 	},
 });
 
+const chatAutoArchiveDaysKey = ["chat-auto-archive-days"] as const;
+
+export const chatAutoArchiveDays = () => ({
+	queryKey: chatAutoArchiveDaysKey,
+	queryFn: () => API.experimental.getChatAutoArchiveDays(),
+});
+
+export const updateChatAutoArchiveDays = (queryClient: QueryClient) => ({
+	mutationFn: API.experimental.updateChatAutoArchiveDays,
+	onSuccess: async () => {
+		await queryClient.invalidateQueries({
+			queryKey: chatAutoArchiveDaysKey,
+		});
+	},
+});
+
 const chatTemplateAllowlistKey = ["chat-template-allowlist"] as const;
 
 export const chatTemplateAllowlist = () => ({