How can I repair a kitchen sink that is leaking from both the handle and the underneath cabinet?
Yes.  The standard repair kit includes two small O rings (and springs) for the ball as well as two (or four) large O rings for the shaft.  You're leaking from both sets of rings I think.

I recommend that you also add a little silicone lube so the ball moves freely on the rings.  Don't use oil or vaseline; they'll dissolve the neoprene rings.

This Danco kit for Delta faucets is typical for single arm sink faucets:

http://www.homedepot.com/p/DANCO-Faucet-Repair-Kit-for-Delta-86970/203194028?N=5yc1vZc6ak