The modules you showed in photos are different. May be i am wrong...
Did you check it in another USB port and another computer?
How much that mouse cost? BTW i am asking it because the module (dongle) price will be somewhere near the new mouse, as it must be imported.
It happened to my roommate in China, he lost module of his new Logitech keyboard (thats quiet expensive)... I searched for it in Taobao (basically Alibaba in China), and found that Logitech doesn't sell components seperately, but there were a few spare part manufacturers who make replacements.. So I bought it for him, and it worked with the help of some 2ndary software.
As we are in Pakistan so replacement would be too difficult to find. I would suggest first check it in another computer. If the same, take it to some electronics engineer who might fix its broken pcb or usb connectors. The easiest way to solve your problem is buy a new one, this brand is not that expensive...
You can also contact A4tech management on this link: https://www.a4tech.com/products.aspx?id=3
I personally like 'silent' mouse and 'mechanical' keyboard for my editing works. You can also go for a gaming or other high speed mouse.