How to get server service to send http requests in Hyper?

main.rs

extern crate tokio_core;
extern crate hyper;
extern crate futures;

use futures::Future;
use futures::future::ok;

use hyper::{Body, Client, Get, StatusCode};
use hyper::client::HttpConnector;
use hyper::header::ContentLength;
use hyper::server::{Http, Request, Response, Service};
use tokio_core::reactor::Core;

pub const INDEX: &'static [u8] = b"Hello world!";

struct MyService {
    client: Client<HttpConnector, Body>,
}

impl MyService {
    pub fn new() -> MyService {
        let core = Core::new().unwrap();
        // This handle should be from the server's event loop but
        // I created a dummy event loop so that it compiles.
        let handle = core.handle();
        let client = Client::new(&handle);
        MyService { client: client }
    }
}

impl Service for MyService {
    type Request = Request;
    type Response = Response;
    type Error = hyper::Error;
    type Future = Box<Future<Item = Self::Response, Error = Self::Error>>;

    fn call(&self, req: Request) -> Self::Future {
        match (req.method(), req.path()) {
            (&Get, "/") => {
                // This route fails and returns an empty response.
                let url = hyper::Url::parse("http://www.google.com").unwrap();
                Box::new(self.client.get(url).and_then(|_| {
                    ok(Response::new()
                        .with_header(ContentLength(INDEX.len() as u64))
                        .with_body(INDEX))
                }))
            }
            (&Get, "/should_work") => {
                // This route properly renders the page.
                Box::new(ok(Response::new()
                    .with_header(ContentLength(INDEX.len() as u64))
                    .with_body(INDEX)))
            }
            _ => Box::new(ok(Response::new().with_status(StatusCode::NotFound))),
        }
    }
}

fn main() {
    let addr = "127.0.0.1:1337".parse().unwrap();
    let server = Http::new().bind(&addr, || Ok(MyService::new())).unwrap();

    let handle = server.handle(); // How to move this handle into MyService?

    server.run().unwrap();
}