A question teachers often ask is regarding helping students learn the skill of one hand playing legato while the other plays staccato. Basically, the student needs to feel that at the moment of playing, both hands go down. The staccato hand needs to release from the keybed while the legato hand stays down.
If this is not enough to solve the issue, I often move to working on the lid. In the example below, Double Agent (From the Music Tree, Book 3), the student can feel that both hands play the first note down, while the left hand stays down and the right hand releases.
If the coordination is an additional problem, I would work with playing whole hands on the lid to feel the larger motor skill first, as a step prior to playing on the lid with the assigned fingering. Identifying (and saying) “Both right both right etc” can be very helpful. I might tap on the lid with the student, have them copy me if right / left is a problem, or if necessary, take their hands and “play” their hands on the lid for them to feel the gross movements.
The student may need to do one bar at a time, repeat, then the next bar, then the two together etc, whatever it takes. At this point, the student might be fine to go straight to the piano, or may need to feel it on the lid actually playing the fingering, with or without exaggerating the staccato movement.
In either working on the lid or at the keyboard, it may be easiest to manage without the rhythm, and I have no qualms in doing so. If the student is overwhelmed, it can be a helpful temporary stage to practice the coordination and fingering without the rhythm. There’s no one set pathway, and this blog is a series of suggestions rather than rigid steps. Some might need all of the suggestions, some might need to stay on one stage much longer than others, and that’s fine.