How do I replace the volume slide?

That really depends on what is broken.

If the plastic part that protrudes from the case is broken, then replacing it is as easy as opening it up and putting in a new plastic piece. This guide walks you through that: Nintendo DS Lite电源开关更换指南

I can not find replacements anywhere except AliExpress. And I cannot find any vendor who still has standalone volume sliders in stock. All I am seeing is full shell kits "with small parts" that include things like the volume slider. For example: https://www.aliexpress.us/item/325680555...

It is possible you could spend some more time looking and find something I cannot right now. This may not be the best option as those available look cheap, and you're paying for a lot that you don't need, shipping can be expensive, and it would likely be a long wait to most countries.

What would no doubt be a better option is finding someone local to you with a 3D printer and having a new switch made custom for you. Many creators will take small projects like that for free or very cheap.

However, if it is not the external switch but the metal one that is on the motherboard, that does require soldering skills. Based on the size and number of contacts, I would rank it as more difficult than soldering something like a through-hole capacitor, but easier than an IC. Definitely not something I would recommend for someone with no soldering skills. However, those switches are cheap and plentiful. For example: https://www.google.com/search?q=ds+lite+...