I have to agree with Flathunter, Sai Kung is going to be your best value family friendly " low density" neighbourhood. We lived in Sai Kung for 3 years, talking from somebody who has lived there, its one of the nicer area's of Hong Kong.
Traditionally Sai Kung appealed to self funded families, who didnt have housing allowances, but its starting to change since some expats are getting squeezed out of HK island due to rent hikes, and moving into the Sai Kung / Hebe Haven / Clearwater Bay area..
You should still be able to get a 2 or 3 bedroom place of around 700FT2, for under $8000.00 per month if you are willing to to live toward Tai Mong Tsai, about 8 minutes by public mini bus or car, from Sai Kung town. Lovely spot, but no longer the best kept secret it once was. Worth it though, as the Sai Kung community is a very close one, and people will know you by name, you arent just another face in the crowd, because its never crowded, although weekends can be busy due to day tripper visitors.
Other area's you might consider if you are trying to achieve affordable family friendly housing, would be Discovery Bay, Tuen Muns Gold Coast and Clear Water Bay. Travel times are going to vary between 45 - 75 minutes to Tai Koo Shing, worth it when you consider your wife/childrens locality happiness, and in turn less pressure on you.
Gold Coast Tuen Mun
Rents for the above will range from $8000 to about $11,000 for upto 800ft2 and 3 bedrooms. Also, like whats been explained to you already, it is extremely unlikely that you will get a lease for less than 13 months. So you will have to budget 13 months rent into your 10.
Hope that gives you a better perspective of whats available to you, with a couple of alternative locations included for good measure.