IoD has a very nice eatery but I dunno whether it's open to Joe Public. Alternatively, if you don't mind a short walk, there's Simpson's in the Strand. Lovely, lovely, lovely!
If you're a reciprocal member (if you were a member you'd know obviously), try the Lansdowne Club, Oxford and Cambridge Club or the National Liberal Club. ahh the IoD - frankly I think it's quite overpriced for the level of service provided.