I think you have two problems, first, the shutoff valves under the sink seem to be leaking (i.e., don't shut off completely). I suggest replacing them with 1/4-turn valves. This may be easy or hard, depending on how they are currently attached to the pipe. If they are compression or threaded on, it's fairly easy. If they are soldered on, then you need some tools and skill. Then, a picture or brand/model of the kitchen faucet would help identify where the problem is. There are thousands of versions in different configurations, so what works for one may not work for the other.