Your dramatic monologue shouldn’t be nearly as long as “My Last Duchess,” but it should have the following criteria: (1) a speaker/character talking, (2) to another person/character who remains silent, (3) at a specific and/or decisive moment in time, and (4) in talking to the other character, the speaker of the poem is revealing his/her character.
Just want to try and help you understand the poetry recreation assignment a dramatic monologue.
Think of a person in your life with a distinct personality–a friend, family member, lover, neighbor, roommate, friend’s boyfriend/girlfriend, the person that serves you coffee, that guy you always see downtown or at the beach or at the gym, etc.–and then just have them talk (monologue) the way they talk (the words they would use and the way they would use the words), but have them talking to someone (and that someone remains silent).
The idea is that as the person is talking, they reveal their character–and their character is probably at least part of the reason why you think they have a distinct personality.
So for example, I have a friend that always is dramatic and pushy and bossy. So I would have her talking to her husband, her husband would remain silent, and I would (hopefully) get across to the reader just with the words she is saying that she is dramatic and pushy and bossy.