Let's pretend the letter is real. Odds are it is not, because how would you look someone up based on they had the same birthday and were born - 20 to 50 years ago - at the same hospital, when you now live across country from each other, and have both possibly married and changed names?
However. If this scenario were real, and you were the other woman, how would you feel if this woman, who came to visit you once before claiming to be writing a paper for school, came back and claimed you were living her life? And THEN, if she had failed to prove she wasn't part of her family she grew up with, you did the testing. It's not cheap to do DNA testing, especially when there is no medical reason that your insurance company will pay for. And then, what if the tests proved you were with the right family all along? Can you say "stalker"?
IF this highly unlikely scenario were real, the ONLY way to proceed would be though an impartial attorney. One who can see about testing the claimant and her family of legal record. THEN, if it is proven she is not related to her matrilineal line, the lawyer can approach the other woman and ask if she is interested in pursuing it. IF she is, he can test her DNA against whatever relatives may have agreed to be tested. If there is a match, he can ask HER relatives of legal record if THEY are willing to undergo testing.
Of course, the claimant should assume financial responsibility for all of this. And at any point along the way, there may be roadblocks. Her relatives may not want to be tested, the other woman may not be interested, the other family may not want to be tested. There may even be no matrilineal relatives in one family or the other to test. Or, she can accept she is who she was brought up to be, and be happy.