You really didn't know how it happened. One day you were taking care of the kid and the next he is attached to you like glue, all you did was take care of him since you are a doctor/nurse.

Killua has never been shown affection and adoration in his life, his parents being awful and (some of) his siblings even worse.

He wasn't going to let you go, not when you were the one that showed him the way of actual family love.