A central creative action in most or all forms of art is transformation, and transformation is highly powerful in music. Just one example of the power of transformation is a fugue, where a relatively short musical statement is transformed many times to create an entire work lasting many minutes.
Usually, once you have the germ of a musical idea, you can generate many related ideas using transformation.
In this case, you have two germs. You have the rhythm that is the rhythm of the melody, and you have the harmonic rhythm of the chord changes. Changing the harmonic rhythm will change how the melody sounds, so that is something you can play with.
To write the rhythms for any instruments that will have a rhythmic contribution, you can base their rhythms on the vocal rhythm. You want to pay attention to how the rhythm instruments accent or diminish the impact of the notes of the vocal rhythm. When choosing what notes to highlight or diminish in impact, consider both the melodic function of the note and the verbal function of the lyrics (if any) surrounding that note.
Pick any one rhythm instrument, write a rhythm for it that has the desired effect with the vocal line, and then have other rhythm instruments double that rhythm or "part out" the rhythm between the instruments. The simplest way to start with the first rhythm is to simply mirror the rhythm of the melody. A simple early transformation is to take one statement of the rhythm of the melody and keep repeating it even when the melody rhythm changes.
Other types of transformation are available, such as rhythmic sequence, retrograde, etc.