It is hard to evaluate your chance to admit to the UK LLM programme unless you have given your approximate overall grades or honour of your law degree.
Btw, what kind of jobs are you aiming at? lawyers? If you want to secure a training contract for solicitor in Hong Kong, I am afraid you have some difficulties if your law degree is not from a common law jurisdiction.

anyway, is Kings famous in Korea?

1. Chances of admission

"Although my background information above is brief, what do you think my chances to be admitted to LSE or other UK lawschools (QMUL, King, etc.)?"

No one can tell you apart from the admissions officer. If one told you 90% and one told you 10%, who would you believe? Confidence is all you need. If you yourself don't believe you can go the school you want, then don't bother applying.

"Would my work experience and attorney license raise the chances?"

It depends on how you present your case in your personal statement and how consistent your proposal of courses is. What is your future career aspiration after the LLM? Why is an LLM necessary for your future development? Is an LLM building up on your existing knowledge or do you simply want to try another career path?

2. Best lawschool for Banking/Finance

"What would be the best choice for banking/finance LLM?"

It doesn't matter when it comes to studying an LLM. If your grades don't allow you to apply for Oxford or Cambridge, then try UCL or LSE. In London you will have more opportunities to network. Schools apart from the 4 named above wouldn't add value to your CV as much as you would think.

"US or UK? My personal aim is to get a job in Singapore or Hong Kong after finishing LLM."

Then why don't you study in HK or Singapore? I have to get back to that question: What is really your future career aspiration?

"I think as my undergraduate is Korean law, postgraduate is US law, it seems a good idea to study in UK to help my education background, but what do you think?"

The diversity can be impressive but may be also indicative of inconsistency. At the end of the day, you'll have to justify your reasons, not only to the admissions officers at law schools, but also to your future employers. Knowing everything means knowing nothing.

"Also, In UK, which lawschool is the best for banking/finance?"

Oxbridge, UCL, LSE. (See above)