Pulling back on the lever isn't supposed to make the wipers move, it only makes the washer nozzles spray. So you're fine on that.
Do you hear the washer pump motor when you pull back on the lever?
If not, then either the motor is dead or it's not getting any power.
If you hear the pump make noise then it is working. Your nozzles are probably clogged, or even the washer fluid pump might be clogged.
At the bottom of the washer fluid tank (behind the fender liner) is the pump. Pull it out of the tank and let the fluid drain out into a bucket or something. Then clean the pump as needed.
To clean the nozzles, first remove the hoses from them on the underside of the hood. Then from the underside of the hood push on the clip prongs built into the nozzle housing. Remove the nozzles from the hood (remember which was left and which was right unless you want to have to re-aim the nozzle jets). Now take a needle and put it in the nozzle jet. Rinse the inside of the nozzle thoroughly.
The reason I had you remove the nozzle from the hood in these instructions is so that you can get all of the sediment/buildup out of the nozzle. If you just insert the needle without rinsing out the nozzle, the buildup will just work its way back into the nozzle jet over time. It would be kind of difficult to rinse out the nozzle while it is still on the hood without getting water all over the place.
Clean out the washer fluid tank so that there is no sediment/buildup in the bottom of the tank.
You should also flush out the hoses with water. To do this just leave the hoses unplugged from the nozzles, plug the pump back into the tank, and then fill up the tank with water. Point the hoses so they spray away from the car and not all over your engine bay... or point them through the engine bay so they spray on the ground underneath. Run the washer pump until you feel satisfied with the flushing. If the pump does not provide adequate pressure to clean the hoses, use a garden hose or something at the end of the hose instead of hooking it to the pump.
Alternatively, use an air compressor to blow out the lines.