The Warburg Institute has no residential accommodation of its own. You may like to consider the following:
Youth Hostel in Euston Road opposite St Pancras station and the British Library. This website also includes lots of cross references to other hostels.
Hotels near the Warburg Institute
Olivelli’s Hotel - from £35.00 per night
It is worth asking about a weekly rate if you are staying long enough but only a few central London hotels offer this.
These are only suggestions. The Warburg Institute and the University of Warwick cannot guarantee suitability or availability of accommodation.