That really sucks - Are those dates in the middle of your Tokyo stay? or at one end?

When shopping around for deals, you can certainly call travel agencies - see what they can offer you.

ww.mytrip.net shows only one place that could accommodate 6people for less than 30,000 on those dates - and it's a backpacker hostel on the East side of town:

http://www.tokyo-backpackers.jp/e/index.html

It would mean a dorm room - six people sharing - 12,600yen TOTAL
(Dirt cheap)

It's fairly close to a Hibiya line station, so you could even get there by subway from the New Sano (if you can handle your luggage on the subway).

You might also call Andon Ryokan - which is also on the cheaper East side of town. They don't have an online reservations service.
Telephone:03-3873-8611

The rooms at the Andon are "appropriately sized" for two people (at 8190 yen) but they will put in an extra bed -- 1 double bed and 1 single bed caters for 3 adults = 11340yen (or 3 single beds)
http://www.andon.co.jp/home/index.html

That will be a squeeze - but probably more civilized than the backpackers place...