Well first of all stay away the North Station District … It has been Sanitised a bit but still remains the Red District of Brussels (quite unsafe at nights and unpleasant at days)

there are quite few swimming pools in Brussels
http://belgium.wikia.com/wiki/Piscines_%C3%A0_Bruxelles

The areas i prefer are 1040 Etterbeek, 1050 Ixelles – Louise, 1000 Center of the city (Sablon, Close to Grand Place etc) , and then further away from the center 1150 Wolluwe, around the Stockel Metro station or in the South watermae boistfort and Uccle etc…

You should avoid North and Midi Train stations areas, Molenbeek, Anderlecht (some parts) Schaaerbeek (some parts) St Josse (some parts) (West – North parts of Brussels and some parts of the Center.

Prices will vary quite a lot depending of the region you go to
I would recommend staying not too far from a metro station as it is a convenient way to get around in Brussels

Try and see which areas with a swimming pool are in areas you might like …

the best site to look for a flat is www.immoweb.be

My experience says that you need to stay a week in a hotel to visit the flats you have spotted (through immoweb) having made appointments with the owners or real estate experts.

best of luck