I went to the New World sauna tonight. I had a fabulous massage. If you go to the 4th floor, go out by the Patio restaurant, make a left and you'll see the sign for it. I was happy with the service. It was NOT the Peninsula, but I didn't feel like going anywhere.
Also, in addition to the hotel salon, the New World shopping center (which is attached to the hotel) has 3-4 including a Toni & Guy salon.
One final, the Intercontinental is right next door so you may want to look at them though it looks like you've got a great spa day booked - enjoy!