To keep someone attentive, active, busy or alert.
on one's toes
Attentive, active, busy or alert.
keep someone busy
To ensure that someone's attention is occupied, often to prevent them …
on the ball
Alert, active, or attentive; on top of things.
tie up
To occupy, detain, keep busy, or delay.
run someone off their feet
To cause someone to become very busy, especially to an exhausting degr …
busy as a bee
Very active; working constantly.