I began teaching music lessons in September of 2001, and I have grown so much as a musician and teacher since then. Sharing the joy of music with my students makes my day each and every day!
The first step for a student to find their rhythm to get them to realize where they are storing the beat in their bodies. Many students have it in their heart or chest area (makes sense with the built-in rhythm of the heart) and many students have it in their head ("head-banging" to a song is actually useful!!).
Once we find this location then we can start making sure that it is "in charge" of the beat and sending it to the hands with some simple clapping exercises at whatever rhythm level they are comfortable with. Sometimes the chest or head will feel interrupted or thrown off the beat by trying to fit into what the hands are doing instead of the other way around. Once students notice this we can then start to correct it and get the beat information going in the proper direction.
Then at this point we add counting of the beats. I like to stick with just counting the numbers of the beat and feeling the places of any notes in the beat (eighth notes, sixteenth notes, etc). I find that counting every single part of the beat makes the student lose a lot of the feel. If they are not sure where the eighth notes are located then subdivide the feeling of the rotation of the head or chest area to find the top and bottom of the beat. Sometimes imagining making a circle out of it will help keep it smooth and continuous.
Once there is a solid beat going and we can get consistent clapping messages to the hands and counting messages from their voice then it is time to try getting the message to the fingers and play the song. Sometimes this step is very challenging, especially if we are working to correct a rhythm the student has already heard themselves play incorrectly many times. Their ears will take over and their chest/head area will try to respond and get interrupted easily. I try to bring them back to focusing on the feel of their beat and having that be in charge.
If they are really stuck in their rhythm the final thing to try would be to do "pretend playing" together on the lid of the piano. So they basically press their fingers on the lid of the piano to the rhythm of the song as if they were playing it. That way they will not be distracted by the sound of their notes and can focus on the rhythm feel. Sometimes doing this together and feeling the beat together is helpful also.