My DSi only works while charging.

Sounds like a battery issue. The DSi is now 17 years old, and even the best batteries would have deteriorated after such a long period of time. Swapping the battery is pretty easy on the DSi, compared to modern devices that features glued down batteries. All you have to do is locate the really obvious battery door on the bottom of the DSi, turn a few screws, and you would have access to the battery, which simply pops out like the one in Nokia phones.
Here's the guide for it:Nintendo DSi battery replacement
There might be corrosion caused by a leaked battery, though, if the battery had remained untouched since you first bought it. In that case wear gloves and remove the battery ASAP. Then use cotton swabs dipped in isopropyl alcohol to throughly clean the whole area. If the issue still persists, you might also consider taking it apart and replace the power board, as there is usually rust on the contacts if the battery has indeed leaked, causing batteries to drain fast. Here's the guide for it:Nintendo DSi power board replacement.
If a new battery and power board didn't solve it, then the issue might be motherboard related, caused by corrosion on somewhere else other than the power board. That is the worst case scenario, and it makes the whole repair much more complicated. Take the motherboard out entirely with this guide:Nintendo DSi motherboard replacement, inspect for signs of corrosion(white stains, sticky stuff, anything out of ordinary), and gently brush that area with a toothbrush dipped in isopropyl alcohol. Hopefully that will get rid of the corrosion. If not, you might consider getting a new motherboard, since repairing the old one would require delicate micro soldering.
But again, it might just be a defective battery.
Get yourself a new battery first and me know if this solved your problem!