@ Jane G, @ Amy S

The Korean name with the closest pronunciation to "Jane" is actually 제인, not 재인 ([ei] vs [æi]).  It is not a traditionally common name (which is probably why it wasn't mentioned in the linked video), but nowadays there seem to be more people with this name (I see 김제인, 이제인, and others on google search).  

I think 재인 sounds like a man's name while 제인 has a more feminine feel.  

Interestingly, the name of the new President of South Korea just sworn in on May 10th has this name - 문재인 (he's male).

He will lead Korea for the next five years.

@ Filipe.

It is a fairly common practice to have a Korean name when foreigners live in Korea.  Of course, many people choose not to, and even when they adopt a Korean name, it is usually just a nickname, not a legal name in most cases.