Jest error:- TypeError: Cannot read properties of null (reading ‘dispatchEvent’)

I am failing to understand the error source in my first attempt at 'jesting'.

import { createElement } from 'lwc';
import Lwcsearchcomponent from 'c/lwcsearchcomponent'
describe('lwcsearchcomponent test suite', ()=>{
afterEach(()=>{
    while (document.body.firstChild){
        document.body.removeChild(document.body.firstChild);}
});
async function flushPromises() {
    return Promise.resolve();
}

it('Input dispatches correctly', async()=>{
const element  = createElement('c-lwcsearchcomponent', {
    is:Lwcsearchcomponent
})
let searchEvent = new CustomEvent('search',{
    detail : { value : 'An' }
});
this.dispatchEvent(searchEvent);

document.body.appendChild(element);
const handler = jest.fn()
element.addEventListener(searchEvent, handler);

const inputElement = element.shadowRoot.querySelector('lightning-input['string-input']')

inputElement.dispatchEvent.searchEvent()
await flushPromises();
expect(handler).toHaveBeenCalled();  
})

The LWC (child) is:

<template>
    <lightning-input 
        type="search" 
        value={searchKeyword}
        onchange={handleChange}
        label={searchLabel}
        placeholder={placeHolder}
        class="string-input"
        is-loading
    > </lightning-input>
</template> 

and

import { LightningElement, api } from 'lwc';
export default class Lwcsearchcomponent extends LightningElement {
@api searchLabel;
@api placeholder;
handleChange(event) {
var keyword = event.target.value;
if ( keyword && keyword.length >= 2 ) {
   let searchEvent = new CustomEvent('search',{detail : { value : keyword }});
        this.dispatchEvent(searchEvent);
        }
    }
}

Error is:

TypeError: Cannot read properties of null (reading 'dispatchEvent')
It dispatches the input value to the parent correctly, but I am trying to conquer testing.

I would be most greatful for guidance.