Interesting question. Your version and the translation have the same meaning but, in my opinion, "não sei escrever cartas" is more straightforward, as in "não sei escrever cartas, nem estou minimamente interessado em aprender (I couldn't care less about learning to write letters, that's it)!", while "não sei como..." makes me believe that the person wants to write letters, but does not know where to start, or doesn't have enough "sensibility" to do so. Could be a subtle way of asking how to do it, but that's just what I think.
So, the translation "I can't write letters" is correct and should be accepted? And another question, then: does the sentence "Eu posso escrever as cartas." have any sense? For example, as the answer to the question: Como vamos felicitar os nossos collegas do Brazil? Or smth. like that:))