What's the onomatopoeia for human cackling?

Question

What's the onomatopoeia for this?

cackle
laugh in a loud, harsh way. "she cackled with laughter"

Like laughing in an harsh, evil way (but that isn't muhaha).

Answer 1

Cackle?

Etymology online says

cackle
early 13c., imitative (see cachinnation); perhaps partly based on Middle Dutch kake "jaw," with frequentative suffix -el (3). Related: Cackled; cackling. As a noun from 1670s...

and

cachinnation
"loud laughter," 1620s, from Latin cachinnationem (nominative cachinnatio) "violent laughter, excessive laughter," noun of action from past participle stem of cachinnare "to laugh immoderately or loudly," of imitative origin. Compare Sanskrit kakhati "laughs," Greek kakhazein "to laugh loudly," Old High German kachazzen, English cackle, Armenian xaxanc'.

Oxford (ODO) says "Middle English: probably from Middle Low German kākelen, partly imitative, reinforced by kāke jaw, cheek."

Answer 2

While some onomatopoeic words may be used as interjections, most interjections do not imitate sounds. Contrarily, onomatopoeic words, such as “buzz” or “boom,” always mimic the noises to which they refer

(http://www.ereadingworksheets.com/figurative-language/poetic-devices/onomatopoeia-examples/)

In 'she cackled with laughter', cackled is the onomatopoeic word.

If you're looking for a more formal sound-word which resembles actual cackling, you can actually just make it up, e.g. ca' ca' ca', she cackled with laughter / hack hack hack, she cackled / ka' ka' ka' she cackled. Just make sure there's context otherwise something like 'hack hack hack' could evoke a very different image!!

Or simply, 'ha ha ha, she cackled with laughter'. Since you're specifying that the laughter is a cackle, the reader will identify ha ha ha in this instance as a cackling laugh.