My sleep [last night] was very broken. (It means it's in small pieces).

That's a tough one. It's not a very natural sounding sentence. One doesn't often talk about sleep like that in the present tense. Maybe it's from a poem? In any case, without knowing the context, I might say in every day language, "I keep waking up," or "I can only get cat naps." More poetically (and with a shift in tense), perhaps, "I fell into a fitful sleep." Oh well, I gave it my best shot. Hopefully that helps. :)