The good news is that the connector is a standard Mini-DIN 6-Pin connector, so it's a widely available spare part. The bad news is that there are no replacement parts available either officially or unofficially. Ever since Saitek was bought by Madcatz which was partially bought by Logitech, official support is essentially non-existent. You'll need to do the soldering yourself or get somebody to help you.

Do the buttons on the base still work, independent of the joystick position? Do the LEDs still light up even when the buttons stop working? If both answers are "yes", it's probably a worn PWM signal cable between the joystick and the baseplate. I'd suggest replacing it with some slightly tougher wire. I've made a guide a while back on imgur, I'm afraid. But it should point you in the right direction. It assumes you know how to solder, though.

There's a few things to troubleshoot. First, does the stick light up as normal when connected? If they do not, it's likely some power cable has been worn out. Next, connect it to a computer (I'm assuming you're running Windows). Find the "Game controller settings" for the X52. You'll find it in Devices & Printers. You'll see a large table of buttons and their current status. What are the state of the buttons on the joystick? Are they all on, all off or flickering? Do they react to being pressed? Next, swap tab to LEDs. Can you change the colour of the LEDs in the stick? If all buttons seem "On" (or flickering) and the LEDs are lit but unresponsive, you may have a damaged PWM cable. It doesn't have to be entirely loose to malfunction, just enough to wiggle and disrupt the sensitive timing of PWM. If so, try out this guide I made a while back on imgur. I'm afraid I haven't had time to move it to this site.