I was in your situation a couple of weeks ago with the same kind of budget and was frantically looking for a short stay apartment. I looked at many possibilities from serviced apartments to shared flats on this site, to estate agent rentals. Unless I missed something, you are not going to find many modern, clean, specialised serviced apartments on the HK side for that kind of money. I was looking at the TST, Jordan area and they were going for at least $8k, and they weren't even self-contained units with kitchen. The only serviced apartments in $6k budget with kitchen were in Tin Shui Wai in New Territories, way up north. The link for that is: www.smart-livingplus.com. They were very nice and hotel-like but only had dial-up internet, plus they were fully booked for a month or two. I also looked around some shared flats in Sheung Wan let by expats but they obviously have compromised freedom and privacy.
I ended up going to estate agents and negotiated a short 5 month lease on a standard apartment with a landlord willing to do it. The standard rental contract is 2 years: 1 year fixed plus 2nd year negotiable, but if you ask, the estate agent will negotiate with landlord on your behalf. Commission is 1/2 month rent, plus 2 months deposit. And with your short lease, they'd probably ask for full payment up front. Offer them split payment to give yourself protection against a negligent landlord if anything goes wrong and they don't do anything about it. I used Centaline estate agents, one of the biggest in HK, so I'd recommend them.
As to internet, HKBN does no contract, pay-as-you-go subscriptions: www.ctinets.com