If you want a non-consecutive rail pass, then the JR East Pass might suit you better. You can use it from Aomori down to the Tokyo area, 5 days of travel during a 14 day period - it can be good to see various sites as well as spend more time at each.
But if you want to go directly from Hokkaido down to Kyoto, why would you waste over 9 hours of a day on a train? You can fly down in about 90 minutes or so from Sapporo or Hakodate, and there are some cheap air fares. Have a look at:
http://www.jetstar.com/jp/en/home
One some days, you can fly Sapporo to Osaka for less than 5,000 yen ($41).
From Osaka you can still use a JR Pass to many nice places, and get your money's worth. Aside from Kyoto, Nara and Himeji are also very worthwhile to visit. Other nice places to see in the area are Iwakuni, Onomichi, Kurashiki, and Okayama. Since you have time, a side trip to Shikoku is also quite possible. Kotohira makes a wonderful day hike:
https://www.youtube.com/watch?v=ley9rVIewoQ
In the Tokyo area, NIkko, Hakone, and Kamakura/Enoshima make excellent day trips.