Ah. That's a mistake in the automation. Definitely de, not di in this case. Let's take this one word at a time:

忙着 Busy / to keep busy doing sthg
自己 Self / ones own.
手 Hand(s)
中的 In the middle of
活 Activity

This translates roughly to: to busily occupy ones hands with whatever work is about, or to take care of whatever chores fall into ones hands. The sentence in the story says that each of the kid's family members are just busily "doing whatever task is at hand".

This character is also pronounced "di" in some cases, but not here.