The verb to type on a keyboard is "pisać na klawiaturze," but I think the response "I am typing/I type" should be accepted since this is an idiomatic verb phrase in Polish to express this. Plus, as a neutral sentence, an English speaker would more likely just say "I'm typing" rather than "I'm typing on a keyboard."
"Piszę klawiaturą"? Well, that would mean that you use the keyboard the way you use a pen (but then it doesn't leave anything on paper)... or possibly that you use like a giant stick to press buttons on a lot larger keybord. Either way, not really probable ;)
Well, it just works exactly like an English: "on a keyboard". You press the buttons that are on the keyboard, so you write on it, in both languages.