If you want a usable lamp on batteries, you'll need a high Lumen per Watt LED which gives the best battery life versus light flux compromise, and also a pleasant color temperature (ie, warm white, not cool white).
All the good lighting LEDs that fit the bill are surface mount only, which is going to be a problem if you want kids to solder them in a workshop... unless you buy pre-assembled PCBs or MCPCBs or strips with the LEDs already soldered.
I think the best would be to use flexible LED strips. They are easy to find online, and the kids should be able to solder them easily, it's just two wires. The strips can be cut to length according to your lamp design (check on the documentation where it can be cut).
Since these strips will use resistors to set the current, this isn't the most efficient solution, but it is a good compromise vs ease of building.
If the strip takes a 5V supply, the easiest is to power it from a cheap cellphone charger. You can also use 4 AA rechargeable batteries, the light will slowly dim as the batteries discharge, but that shouldn't be a problem.
Forget about the 9V battery. These are only for low current applications, also they are extremely expensive.
Another option would be to use some hig-power LEDs, which you would buy on star MCPCBs, but you will need a switching driver. These aren't expensive, but it's still an extra part. You can try aliexpress or banggood, there are tons of options.