At first sight, the modern, if Art Deco influenced, ambience of Haxells restaurant, within the Strand Palace Hotel, does not suggest Old London. But there is an interesting backstory. The hotel itself dates back to 1909, but took its present form in the 1920s, with a redesign in Art Deco style. The expanded hotel incorporated the adjoining Haxell’s Family Hotel, from which the restaurant takes its name. Alas, most the Art Deco features disappeared over the years, with the V&A museum taking some of them, though the restaurant decor gives some reminder of past glories.
For those interested in information history, the hotel holds specific interest. It was here that Leo Szilard, the Czech physicist who first formulated a relation between information and entropy, stayed during his time in London, and from which he set out on the famous walk on which he suddenly made sense of the nuclear fission problem, while waiting for the traffic lights to change so could cross nearby Southampton Row.
As for the restaurant itself, it is a very useful standby, should you find yourself among the crowds in Covent Garden or the Strand without a reservation. The modern British food is good, without being spectacular; Leo Szilard would have liked it that way. Given his interest in biology, I expect he would have been intrigued by the superfood salad.
The Occasional Informationist 
Never knew that. Useful tip, thanks David.