Add SQLite connectivity check to healthcheck endpoint

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

app/api/health/route.ts

@@ -1,5 +1,12 @@
 import { NextResponse } from "next/server";
+import { client } from "@/lib/db/client";
 
-export function GET() {
-  return NextResponse.json({ status: "ok" });
+export async function GET() {
+  try {
+    await client.execute("SELECT 1");
+
+    return NextResponse.json({ status: "healthy" }, { status: 200 });
+  } catch {
+    return NextResponse.json({ status: "unhealthy" }, { status: 503 });
+  }
 }