99% of people.
a)Buy a replacement case. There are ones that are from the factory/from a "third"/ghost shift, ones that are from the factory but failed quality control at some level and ones that are made by someone that found a machine and put the effort into making one that sort of works. Reviews if you can find them might sort this.
b) Cannibalise an existing DS lite to sort it. I was buying them for next to nothing last year when we still had car boot sales and whatnot so there is that.
Others. You can fix it. You will need to fully disassemble it as trying to glue things together is tricky* and you don't want to mess with the wires and whatnot inside it. https://www.popularmechanics.com/science/tips/a584/2569841/
You might also melt some metal into the joints but that is harder still.
*tricky in this case means you will need to cut shims of suitable material (not all plastics work with all solvents/glues, take advantage of that) and dimensions that can go between things and protect it and still be delicate in application. Probably still also finish it with a needle file.
I would not say it is hard. Just that it is tedious.
That link. Could go either way from where I sit. If they are legit then most likely they did b) above*, sold off all the buttons, screens, slots, motherboards and whatnot separately, and looking at the rest of their listings they are doing just that (that and the fuses for things likely being them buying in a reel, the picture there displays some cut tape from one, for fractions of a cent each and slicing them up to send out to mark up many more times -- the claim they are a student so likely someone in the electrical engineering persuasion as tend to be the ones to play such games/figure such things out).
*if you have the time and funds to sit on things then something is often worth more as parts than it is as a whole. Sentimentality and lack of funds lead people to pay inflated prices to fix things.
Lets be honest, the seller you have linked to is charging extortionate prices,
Each time I re shell a DS I clean up the old parts that were not broken and keep them, sometimes I'll put an LCD in and sell the top half "ready to go" as a lot of people who need to replace the screen often have dead pixels or a dodgy flex cable, even then I sell them for less than the price of his Slot 2 covers,
If you still require the parts, I'd suggest looking for a UK seller if it's original parts you want as they won't charge the shocking prices he's charging.