test: add xUnit test coverage for API controllers and client services

Create ChatAgent.Api.Tests and ChatAgent.Client.Tests projects with xUnit
and Moq. Test HealthController (200 + valid response), ChatController
(SSE streaming with mocked upstream, error handling), and ChatApiClient
(delta parsing, error events, health endpoint). 6 tests, all passing.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

tests/ChatAgent.Api.Tests/HealthControllerTests.cs

@@ -0,0 +1,29 @@
+using System.Net;
+using System.Net.Http.Json;
+using ChatAgent.Shared.Models;
+using Microsoft.AspNetCore.Mvc.Testing;
+
+namespace ChatAgent.Api.Tests;
+
+public class HealthControllerTests : IClassFixture<WebApplicationFactory<Program>>
+{
+    private readonly HttpClient _client;
+
+    public HealthControllerTests(WebApplicationFactory<Program> factory)
+    {
+        _client = factory.CreateClient();
+    }
+
+    [Fact]
+    public async Task GetHealth_Returns200_WithValidHealthResponse()
+    {
+        var response = await _client.GetAsync("/api/health");
+
+        Assert.Equal(HttpStatusCode.OK, response.StatusCode);
+
+        var health = await response.Content.ReadFromJsonAsync<HealthResponse>();
+        Assert.NotNull(health);
+        Assert.Equal("healthy", health.Status);
+        Assert.True(health.Timestamp > DateTime.MinValue);
+    }
+}