Two reasons I can think of:
- Culverton asked her to, because he wanted Sherlock to make him more famous, with his accusation of being a serial killer:
Culverton gave me Faith’s original note. (She stands up.) A mutual friend put us in touch.
There would be no reason for Culverton to give Faith's note to anyone, unless he wanted to play with Sherlock.
- Eurus (in the guise of Faith) wanted to test Sherlock. From the same speech by Eurus
I added some deductions for Sherlock. He was... quite good. But... he didn't get the big one.
Not only does she want to test Sherlock's deductions, she wants him to see the "big one":
