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Information concerning hostels can be found further down the page.
One possibility is Hotel Arcadia which is located very near KTH (KTH is approximately at the Östra station):
One very good site for booking hostels can be found at Kilroy travels. (Look at the right side of the page, near flight booking possibilities.)
We have not tried the following hostels, but their price/location/quality seemed good (or when it comes to location, it was sometimes rather more interesting than exactly optimal for getting to KTH). It is always worth checking from the webpage whether sheets/towel/breakfast/Internet/etc is included (I haven't detailed everything here).
This looks fairly reasonably priced and the location is quite good: City backpackers.
Belman Vandrarhem, Stockholm, Sweden (doesn't have a webpage) looks reasonably priced.
This one is an old jail but nowadays an award winning hostel (see, you can get there legally, nowadays): Långholmen Hostel STF/HI.
Prices in huge rooms are cheap and the location is in the centrum of the city, near the railway station and Cityterminalen: City Lodge Stockholm.
This one has excellent location (and cheap prices): 2kronor Hostel - Vasastan.
Great location, breakfast is included in the price: Hostel Bed and Breakfast.
Three different locations within the city (all good for tourists), room prices cheap: Best Hostel.
Good location for a tourist, cheap prices for private rooms: Archipelago Hostel Old Town - Stockholm.