Word or expression for the noise in a busy house with kids?

Words like racket and din could lead you to other synonyms. Lexico has

racket²
1 A loud unpleasant noise; a din.
the kids were making a racket

Another suggestion is, from Lexico

hubbub
1 A chaotic din caused by a crowd of people.
1.1 busy, noisy situation.

You could say

Our house is always a hubbub of activity.

Cacophony (Noun): A harsh discordant mixture of sounds.

"a cacophony of deafening alarm bells" Lexico

The word ruckus can be used. Google gives the following definition and provides an example with a child:

a disturbance or commotion.

"a child is raising a ruckus in class"

Clamour (Noun): A loud and confused noise, especially that of people shouting.

Example: From behind the group, a great clamour arose — Lexico

_{It can also be used as a verb.}

Or tumult.

Tumult (noun): A loud, confused noise, especially one caused by a large mass of people — Lexico

Example: a tumult of shouting and screaming broke out.

It's a bit more dramatic, but "pandemonium" (Noun) comes to mind.

"Wild and noisy disorder or confusion; uproar." Lexico