One thing to keep in mind is that even in "expensive" cities like HKG, weekly/monthly rentals may not be so bad- check out sites like vrbo.com or airbnb.com. And if your board is taken care of in HKG, the rest of the city doesn't cost as much, especially transportation.

I haven't been to KL, but I think BKK, HKG or SIN would all be great bases. I guess it depends on what you want out of a city- SIN is the cleanest with the most English, HKG is dense but modern, and BKK is crowded, noisy, and full of things to discover. Obviously there is more to each city, but the point is you need to figure out what you want out of a home base. All three cities don't have more than a week's worth of things to see, but living there adds a whole other dimension, and I obviously BKK is the easiet gateway to the rest of Thailand, which you could easily spend a month or more exploring.