Congrats on your upcoming move! I second evgreen on Stanley/Tai Tam as a lovely place to live.
In all honesty, everyone has different requirements, preferences and deal breakers with regards to areas to live. It's hard to gauge from afar, so take your time to find the right area. Suggest a serviced apartment for the first few weeks to allow yourself some time to check the various areas out.
I find that a good property agent makes a huge difference - not to generalize but a newcomer to HK is often shocked at the state of HK rental apartments ( they can be very old and basic, circa 1970s or cookie cutter tiny in newer
developments) so finding the right agent to show you round makes things easier.
Take a look at Habitat-Property.com - they cater to the Expat market and focus on a few select areas in HK. The budget that you've indicated throws up quite a few nice options in Stanley, so worth giving them a shout. And no, I'm not affiliated in any way to Habitat! Just been here long enough and find their services useful and spot on for "a typical Expat".
re Stanley/Tai Tam - it's on the Southside of the island so very low density, great sea views, beaches and a very relaxed way of life compared to the hustle and bustle that is HK. Great for families as a result.
Getting to Quarry Bay is not bad as you're going against traffic, although yes, if you're trying to get to Central, I can see why others would consider it a distance but it's still only at best 40mins in lousy traffic. And coming from London, i guess that is not so bad, no?
All the best for your move!