DK / Nintendo Joysticks are actually made very well. All metal construction, so the only thing that can really wear out is the spring.
There are 5 pieces to a DK Joystick.
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1) Metal shaft with black ball top
2)Upper bit
3)Spring
4)Lower bit
5)Retaining clip
You will have to take the joystick apart, starting with the retaining clip on the bottom of the joystick. Be careful as this is under tension and once the clip is removed the spring and retaining bits can fly off and you could lose them. Once the retaining clips is removed you can then removed the spring and upper and lower bits from the bottom and pull the joystick out from the top.
Now sand every piece down with sandpaper except the spring. Make sure there is no rust and they are all nice and smooth.
Now examine the spring and judge if it is toasted or can be reused.
If spring it toasted you will have to buy another from mikesarcade.com or someone on these forums might have an extra. If the spring can be reused just put all the parts back on in opposite order and from here the joystick should work perfectly.