Finding a file extension in a string using shell script

I have a long string, which contains a filename somewhere in it. I want to return just the filename.

How can I do this in a shell script, i.e. using sed, awk etc?

The following works in python, but I need it to work in a shell script.

import re

def find_filename(string, match):
    string_list = string.split()
    match_list = []
    for word in string_list:
        if match in word:
            match_list.append(word)
    #remove any characters after file extension
    fullfilename = match_list[0][:-1]
    #get just the filename without full directory
    justfilename = fullfilename.split("/")
    return justfilename[-1]


mystr = "the string contains a lot of irrelevant information and then a filename: /home/test/this_filename.txt: and then more irrelevant info"
file_ext = ".txt"

filename =  find_filename(mystr, file_ext)
print(filename)

this_filename.txt

EDIT adding shell script requirement

I would call shell script like this:

./test.sh "the string contains a lot of irrelevant information and then a filename: /home/test/this_filename.txt: and then more irrelevant info" ".txt"

test.sh

#!/bin/bash

longstring=$1
fileext=$2
echo $longstring
echo $fileext