"Could you clean up the room?"
Grammatically, the 把 construction requires a compliment after the verb (at least, that's what I was taught.) So you can't end the sentence with 打扫. You need to put some kind of complement after it. (I'm pretty sure 吗 does not count as a complement.) A complement is anything that gets tacked on directly after the verb in Chinese that further explains, specifies, or otherwise modifies it.
打扫 is the verb "to clean" 干净 is the adjective "clean" (describes a thing that is clean.)
In 打扫干净, 干净 is a resultative complement. It describes the result of 打扫. 打扫干净 means "to clean until the thing is entirely clean," as opposed to cleaning for 5 minutes then stopping to watch a movie.
I believe that Duo translates 打扫干净 to "clean up" (instead of just clean) because "clean up" has a similar sense of completing the cleaning.
Using the 把 structure in this case is certainly correct. I'm honestly not sure whether your sentence is also correct or not. However, the 把 sentence sounds more natural to me.
This is just how I personally think about it, but I think of the 把 structure as a solution to this situation where you have two things that ought to go right after the verb. Using 把 gives you another place to put the object. It sort of unclutters the end of the sentence.