You can't find a service apartment in the Central/Wanchai/North Point area for $7K, apart from the W Studios room, which you will not like. I think you have to increase your budget to $8 to $10K.
Service Apartments come with hidden costs. I discovered that I had to pay additional for broadband ($500), electricity ($1K), water ($200), stamp fees ($1K) and laundry ($40 per wash). These add on costs are non-negotiable and another source of income for them.
Since you are on a budget, I suggest you take a $5 - 6K leased flat (6-month contract) instead cos it will be cheaper and larger. You can also manage your own utility and phone bills.
To find a flat use the GoHome search engine.
There, I've done one for you:
Once you get here just stay in a hotel or hostel while you look for an apartment.
Don't be too fussy and you'll find your flat either immediately or within days.