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.