Figure out exactly what type of British accent your character has, then find an actress (or we can help you with that) who has the appropriate accent, then listen and repeat, listen and repeat. Also write out the sound differences phonetically how they sound to you.
I’m not sure which part of England this comes from, and I’ve heard it from Brits as well as Aussies, but here is a good example: In the U.S. we might say, “I don’t know.” Whereas with certain British and Aussie accents the phrase sounds like, “Uh die yunt nigh.”
Another example comes from my favorite Brit, Charles Shaughnessy (aka Maxwell Sheffield), as well as Daniel Ratcliffe (Harry Potter). In the U.S. we say the word “anything” but these two fine fellows pronounce the word as “enathin.”
So find out where your gal is from, find a British actress (even one from a documentary) who speaks the way your character would, then listen to and repeat phrases, over and over again. After you’ve done that for awhile, have your friends ask you random questions and try to reproduce the accent while you answer.
For some reason, I’ve always been able to imitate accents, much to the amusement of my friends and relatives. I do a great Bill Clinton as well a credible Liverpudlian accent.