John Lawler is correct, in the context of ordinary conversation (though "what she would like to have" is awfully stiff for chatting with a friend).
That said, "so" indicates that you bought the purse because you didn't know what she might like (at the time you bought the purse). The relevant action of knowing (or not knowing) has to come before the buying, and therefore you should use the past tense.
If you prefer the present tense, then a good option might be: "I really don't know what she would like to have for a birthday present, but I bought a purse."
Stylistically, though, in ordinary conversation I'd say either, "I didn't know what to get her for her birthday, so I bought her a purse," or "I don't know what she wants for her birthday, but I bought her a purse."