Not so much a recomendation, but maybe one to avoid... DH and I stayed at the Coogee Sands Apartments, and to be honest, it wasn't exactly what I was after... we stayed there twice - the night before we flew out from Sydney airport, and the night that we came back. We got upgraded to a suite the first night we were there - little living room, kitchenette, seperate bedroom and a huge balcony... not bad, but I don't think there was a washing machine. On the way back, we knew we would be there most of the day as well as overnight, so we booked an ocean view room. We were on the front of the building facing the beach, but between the trees and the powerline, we couldn't see a thing. It was a standard hotel room size, but did have a small kitchenette and a tiny balcony with a wrought iron table and chairs (not comfortable at all).
The big problem for us (and I think it would be an issue for you with twins and probably prams / bags etc) was the layout of the hotel. DH and I had gone away for a scuba diving holiday and had taken all our gear with us, so as well as our normal luggage, we also had about 40kg worth of dive gear. The hotel is split level and the lift doesn't go to all levels. The car park is on two levels and reception is in the middle, so no matter where you park, you either have to walk up or downstairs. Then to get up to the room we had to get the lift up about two floors, then walk up a flight of stairs, up a long corridor, then up a smaller flight of stairs. The stairs are also quite narrow, so it would be hard to navigate it with a pram and kids in tow.
I remember when we were looking at places to stay around Coogee, there wasn't a huge amount to choose from... might be worth your while to look at Stayz.com.au
to see if you could rent an apartment for a few days instead?