How do I add custom IChatClient middleware to chat client registered with a Semantic Kernel?

I'm trying to figure out how to add middleware to a chat client added to the Kernel. It seems like I need the actual instance of the chat client in order to add the middleware?

This is the documentation for the chat client middleware https://learn.microsoft.com/en-us/dotnet/ai/microsoft-extensions-ai#custom-ichatclient-middleware

This is my setup

services.AddKeyedTransient<Kernel>("Kernel", (sp, _) =>
{
    ...
    var bedrockRuntime = new AmazonBedrockRuntimeClient(awsCredentials, clientConfig);

    var kernelBuilder = Kernel.CreateBuilder();
    kernelBuilder.AddBedrockChatClient(
        modelId: awsBedrockOptions.GetModel(LlmOptions.ModelPurpose.General),
        bedrockRuntime: bedrockRuntime
    );

    return kernelBuilder.Build();
});

This is the middleware:

public sealed class LlmLoggerChatClient(
    IChatClient innerClient,
    LlmLogger llmLogger)
    : DelegatingChatClient(innerClient)
{
    public override async Task<ChatResponse> GetResponseAsync(
        IEnumerable<ChatMessage> messages,
        ChatOptions? options = null,
        CancellationToken cancellationToken = default)
    {

        var response = await base.GetResponseAsync(messages, options, cancellationToken);
        await llmLogger.LogAsync(...)
        return response;
    }

    public override async IAsyncEnumerable<ChatResponseUpdate> GetStreamingResponseAsync(
        IEnumerable<ChatMessage> messages,
        ChatOptions? options = null,
        [EnumeratorCancellation] CancellationToken cancellationToken = default)
    {
        await foreach (var update in base.GetStreamingResponseAsync(messages, options, cancellationToken))
        {
            yield return update;
        }
    }
}

And this is how I finally use the kernel

ChatCompletionAgent agent = new()
{
    Name = "Search Agent",
    Kernel = kernel, // Injected via DI
    ...
};

var response = await agent.InvokeAsync(...

At what point should I/can I extend the Bedrock chat client?