The Strand Palace Hotel has been recommended before by several AB contributors who've stayed there. To get to the theatre from there, it's a half mile walk (along the Strand and across Trafalgar Square):
http://www.strandpalacehotel.co.uk/
(If you'd rather not walk, bus services 6, 9, 13, 23 and 139 combine to provide a really frequent service for the journey)
The nearest hotel to the theatre is probably the RoyalTrafalgar:
http://www.thistle.com/hotels/trafalgar/
Chris