Who should pay for the falling lamp?

If she did something wrong, something she should have known would cause the lamp to fall, she should pay. But you didn't give a reason why the lamp fell. All you said is that it fell after she cleaned it. That does not mean she caused it to fall. It's probably just coincidence. Because one thing happens after another thing does not mean the first thing caused the second thing.

As she cleaned the lamp with your permission it's down to you.

It is yours. In particular since you were not paying her, and she was helping you. You obviously did not have it securely fastened so that it could fall. The friendship is worth far more than the light.

Unless she was hanging onto the lamp and swinging on it, you should pay. 
Lamps are supposed to be secure enough for anyone to clean them. Lamps are not supposed to just fall while someone is cleaning them. 

No ones fault, it was just either old or not properly secured. The homeowner is responsible for upkeep and maintaining a safe enviroment. The cost should fall on the homeowner as well as any damage to any other property or person.

Tip - DON'T help her with her homework. Especially English.

If you were aware that she intended to clean the lamp and said nothing to object and were not paying her to clean the lamp it is unlikely she is legally obligated to pay for the lamp. You don't say if she was at fault or if some part of the lamp failed causing the lamp to release, fall and break. If that's the case even a professional would not likely be held responsible. On top of that, the fact that the lamps were old very likely means that their value (used) is very low if they had any value at all.

I would demand Alvin & The Chipmunks pay.

And YOU can't figure this out yourselves?
No hope for the BOTH OF YOU.