If you have a repair kit: https://www.amazon.com/dp/B07G122BVS/ref=cm_sw_r_cp_apa_i_OHgmFbBS79H1Y
You can take apart the controller case and see what's wrong with the joystick.
I'd recommend watching tutorials on YouTube to see how to take apart the controller. You might just need to slide it back into place, or you might have to re-solder it, or buy a new PCB board off eBay and replace the one in the controller.
I'm a bit late but I have a pair of xbox sticks in mine, they work perfectly, not sure about elites though
The modules use the same screw as an xbox one controller IIRC this is the toolkit I have
Nope. I used this one for the first 4 screws and it worked just fine but the fifth one was just faulty https://www.amazon.com/dp/B07G122BVS?ref=ppx_pop_mob_ap_share
Watch a few disassembly videos, I found it quite easy to take all my controllers apart thanks to those videos, you will however need a tool kit, something like this should do the trick although I have a kit similar to this here in the UK.
Yeah no problem! Between this set of Torx screwdrivers and this set of Philips and flatheads I'm pretty good on any small electronics I would need to work on, plus it was cheaper than bigger sets 😅