Here's a step-by-step, best I can do. I will make a diagram too if I can figure out a way to upload it.
Get 4 of these...
Each one has a red wire and a black wire.
You'll also need about a foot of additional wire- red or black, up to you. Let's assume black.
You'll also need a roll of electrical tape.
Using wire strippers if you have them, or a knife if you don't, strip back maybe an inch worth of insulation on each wire.
Put all the battery holders in a row so the red wires are always on the right and black wires on the left.
Twist the red and black wires together everywhere where there are two holders next to each other. You'll end up with a black wire by itself, 3 pairs of red+black twisted together, then a red wire by itself.
Take that extra foot of black wire and use it to extend the black wire at the beginning of the chain of battery holders.
Use electrical tape to cover up all these places that you've twisted wires together.
Now, you should be able to put batteries into the holders, open up the battery compartment for your LED lights, and touch the red wire to first flat plate and the the black wire to the first spring. You'll have to experiment to figure out which of the plates and springs are actually connected to the LED's, but you can safely mess around with live 6V power. Just touch wires to battery contacts with the LED's turned on until you see them light up. Then you'll know you have the wiring right so far.
Next step, rip the battery compartment apart. You'll probably end up needing a saw to cut the plastic away for the unused portion. Hopefully, the plate and spring that actually hook up too the LED's will be on wires, and you can just strip back those wires and twist the metal together like you did before. If they're part of a circuit board or something, you probably will need a soldering iron, but it's still totally possible.
Once you've got this nice loose heap of parts all spliced together and functioning, just figure out a way to secure it all together in a slim fashion and slide it into your staff. Electrical tape, maybe some hot glue or superglue, maybe a long piece of plastic to glue everything to- all these things will be your friends when figuring out how to hold it all together. Just try to make sure there's no tension on those spliced wires, and that you can still replace the batteries when they run out.
Sorry this is so long winded, I was trying to explain it as simply as possible.
I probably should have drawn a picture FIRST.