Keyboard Backlight Connector

If it's just the little folding retainer clip that broke off, you may be able to cobble together a solution. If you can buy a salvaged motherboard for parts, you can take the clip from it to use on your socket. This video shows how.

Some people have been able to connect the ribbon cable to the socket without a clip by taping the cable in place with electrical tape or hear-resistant tape.

If the repair shop broke it, why not ask them to fix it?

Whenever touching components or working inside a computer, wear a grounded wrist strap, also called anti-static wrist strap, ESD wrist strap, or ground bracelet. I know many people do such work without this, but it's a cheap and sensible precaution. Rest the laptop on an anti-static mat or at least a reasonable alternative such as corrugated cardboard.

I have not trusted them so I will not back to the same shop to damage other thing

You're not going to be able to find that part short of purchasing an otherwise dead board from the same or a similar model, and salvaging the component -- and even then, you will need the equipment to do surface mount soldering.  

More likely, you will need to replace the system board if your need a functional backlight.  

I will search for a dead board thanks ♥

Hi @Yasser Saied welcome to this free user-to-user Alienware laptop discussion forum. 

The cable socket part that was broken continues to be your property and the repair shop is obligated to return it to you. You should be able to make a claim on their insurance. 

Please contact $100 Dell Laptop Repair | Dell Repair | Laptop Repair | Notebook Service Center (parts-people.com) to determine replacement cable socket captivation part availability. The remainder of the socket looks good in your image, so just insert the missing part.