What is the first name and is it significant? If they chose a name in honor of a relative on either side, I don't see the big deal. I know friends who have a bunch of people in the family named "Joe" after someone in the family.
I kind of don't understand the point of asking permission if you're just going to go through with what planned on anyway. That seems stupid.
If you're game to be peace maker, I'd suggest that the new child goes by her middle name. I have a friend who is the fifth "Mary V." The V is always some woman's name beginning with a V, ie. Mary Victoria, Mary Viola, etc. No one actually goes by there first na,e.
Does the adopted child know you call her by the middle name and not her new first name? Are the parents cool with that? That's off-topic, but I think my parents (I'm adopted) wouldn't have been fine with that, if I had ever gotten in touch with my birth parents.
I personally wouldn't care if my birth name were recycled in that family. I feel no connection to it, and I am who I am now, not the person with the Korean name. It's just a name, but to me, I haven't been that person since I was a baby. I have an odd take on being adopted, though, since I have no desire to "get in touch with my culture" nor do I care to see my birth parents.