The letter makes her cry. It makes her cry before she's read more than one line. Once upon a time she knew Sirius' hand-writing perfectly, second only to her own and her husband's. Once he would have addressed her as Medea, a childhood nickname that he had refused to stop using, even after she outgrew the bitterness at her namesake, even after she became a mother. She'd always hexed him when he said it in front of Nymphadora who, despite the name, she'd prayed would grow up as little Black as possible.
One wish granted, then. She's certainly all Tonks.
Another wish granted; somehow Sirius knows that she still loves him, that she knows he didn't do what they said he did. She can't imagine how – she knows better than to think that Azkaban would have let him keep how close they'd once been. She knows that she has a starring role in several of his happy memories. Maybe not the happiest – she'd seen him around James Potter enough to accept that – but happy, happy days.
And he remembers enough to know that she loves him, and, judging by the letter, he loves her back. A miracle.