Hey unfortunately Sydney and cheap dont go together at all.
Its damn exie but its a great place!
All the places we have stayed at didnt have pools sorry (I dont think a lot do there) my suggestion would be to look on Wotif and then put in pool as the search and then contact the hotels directly to see what their priceing is - because wotif is only 3 months in advance if that makes sense.
I LOVED LOVED LOVED Sydney the things we did were:
*Went on the Ferry to Luna Park at night (fun park amazing lights on the bridge etc but probably couldnt do with a child?)
* Went on Ferry to Manly (takes about 20 minutes) and its just a nice wee place there we went over for lunch and had a walk along the beach.
* Markets on the weekends at the Rocks
*Sydney Aquarium is amazing really is - but SO expensive! But it was great. The sharks where amazing. Although Ive never been to Kelly Tarltons or whatever its called so maybe just as good as there?
* Go to Bondi Junction (where the train stops) they have THE most amazing mall there, I mean the best one ever. Its about 8 stories high - and I class myself as a shopaholic - and this mall was so overwhelming for me and I said to DH that I was overshopped - he couldnt believe it! Haha.
* The gardens at Darling Harbour are good and there is playgrounds there etc. Was great we had lunch there a couple of times on the banks was really nice.
The best thing we did was buy a 7 day train and water pass (even though we were only there for 3 days) it was great got us around everywhere we needed to go and that pass included going to Manly and to Luna Park etc. Every night we went to Luna Park and back just to go on the boat at night because it was so amazing! It worked out very cheap.
Enjoy yourselves you lucky things, Im so jealous I would love to go back to Sydney again!