How do I add WebSocket with database connection to ASP.NET application?

Question

I have a regular ASP.NET app, which uses routes and etc. I also want to add WebSockets with database connections to the app, and have come to the conclusion middleware is the best way to do this, but I can not get it to work with the database. This is currently my WebSocketMiddleware class:

public class WebSocketMiddleware
    {
        private readonly RequestDelegate next;
        private readonly DatabaseService databaseService;

        public WebSocketMiddleware(RequestDelegate next, DatabaseService databaseService)
        {
            this.next = next;
            this.databaseService = databaseService;
        }

        public async Task Invoke(HttpContext context)
        {
            if (context.WebSockets.IsWebSocketRequest)
            {
                WebSocket webSocket = await context.WebSockets.AcceptWebSocketAsync();
                await HandleWebSocketConnection(context, webSocket);
            } else
            {
                Console.WriteLine("Not ws");
                await next(context);
            }
        }

and this is the Program:

var builder = WebApplication.CreateBuilder(args);

builder.Services.AddControllers();
builder.Services.AddEndpointsApiExplorer();
builder.Services.AddSwaggerGen();

builder.Services.AddDbContext<DatabaseContext>(options =>
    options.UseNpgsql("address"));

builder.Services.AddScoped<DatabaseService>();
builder.Services.AddScoped<WebSocketMiddleware>();

builder.Services.AddCors(options =>
{
    options.AddPolicy("AllowLocalhost",
        builder =>
        {
            builder.WithOrigins("https://localhost:3000")
                .AllowAnyHeader()
                .AllowCredentials()
                .AllowAnyMethod();
        });
});

var app = builder.Build();

app.UseCors("AllowLocalhost");

app.UseWebSockets();
app.UseMiddleware<WebSocketMiddleware>();

app.UseHttpsRedirection();
app.UseAuthorization();
app.MapControllers();

app.Run();

And this is the DatabaseService I am attempting to inject:

public class DatabaseService
    {
        private readonly DatabaseContext databaseContext;

        public DatabaseService(DatabaseContext databaseContext)
        {
            this.databaseContext = databaseContext;
        }

// methods //
}

I have tried removing and adding WebSocketMiddleware as AddScoped, but both fails from their own reasons. If I remove the DatabaseService from the middleware, it works as expected.

Errors: When adding scoped: 'Unable to resolve service for type Http.RequestDelegate while attempting to activate WebSocketMiddleware'

When removing scoped: 'Cannot resolve scoped service DatabaseService from root provider'

If I remove RequestDelegate from the middleware, I get this error: 'A suitable constructor for type WebSocketMiddleware could not be located'