What could be causing my kitchen faucet to not work?
The diverter for your sprayer is faulty. Typically, when the sprayer is used, the faucet will completely stop. Once you stop using the sprayer, the diverter redirects the flow back to the faucet. Depending on your faucet model, this part may be replaceable, and you may be able to open up the valve and find something clogged or broken inside. But I'd personally take the opportunity to replace the entire faucet. Pull-out faucets eliminate this valve, look cleaner, and leave the old sprayer hole available for a soap dispenser.