I agree, there aren't a whole lot of options for renting a house in the bursa/iznik area on the internet. But I would recommend that you search for accommodation in the marmara sea are in general- you are bound to find something near that you like. For this there are a ton of sites you can look into. You said you were looking for a house, but it seems like there are more options for apartments. It looks like though you can find some pretty large apartments. Here is a site with a lot of listings:
http://www.vacations-abroad.com/turkey/marmara/default.aspxHere is another site with apartments, but they also list interesting hotels if you want to at least consider that as well! If you look at the descriptions of the hotels some of them mention that they have spa and wellness facilities, which sounds like is what your wife is looking for. Some of the apartment options also have swimming pools :
http://www.casamundo.co.uk/turkey/marmara_sea/And if you are definetely set on renting a house or villa, here is a site, although the villas might not exactly be in the region you were intially looking for:
http://www.turkeyholidayvillas.co.uk/Hope this helps anyone else looking to go to the marmara sea region!