For that budget you're limited on where you can go. Most of the "big" serviced apartments (Shama, etc...) are more in $20K+ per month range.
I've seen online listings for a lower-cost serviced apartments in Happy Valley/Causeway Bay but I've never visited and don't know how valid the $$ listings are. Anyway you can give these a try. They should all be relatively close to Jockey Club. Maybe others on this forum can comment on quality?
H studios
41 Yik Yam st. Happy Valley
Rent Range: HK$8,000 - $10,000/month
Tel: (852) 2815 6866
Fax: (852) 2815 6768
E-mail: [email protected]
De Victoria Serviced Apartments
9 Yee Wo St., Causeway Bay, Hong Kong
Rent Range: HK$6,800 - $17,000/month
Tel: 2890 8810, 9094 8994
Email: [email protected]
Soho Apartments
47 Paterson St., Causeway Bay, Hong Kong
Rent Range: HK$3,800 - $9,800/month
Unit Size: 200 - 350 sqft.
Tel: (852) 2392 6868
Fax: (852) 2576 7509
E-mail: [email protected]
Cozy Apartments
CWB and Wanchai
Rent Range: HK$6,000 - $9,000/month
Tel: (852) 2836 7788
Fax: (852) 2910 7000
E-mail: [email protected]