EytibiEytibi is right, but for me it sounds more like playing with the dog. If you want to say 'The kid took care of the dog' in most cases you would use "Das Kind hat sich um den Hund gekümmert".
To be exact:
'The kid was busy with taking care of the dog.' = "Das Kind war damit beschäftigt, sich um den Hund zu kümmern."
'The kid was busy in playing with the dog.' = "Das Kind war damit beschäftigt, mit dem Hund zu spielen."