The rink is likely to be the the London Ice Club, which now does not exist anymore. According to The Times (26 January 1933), Mollie Phillips took part in a championship there. 

 

The London Ice Club was opened with a special dinner on 14 January 1927. It had been conceived by the enthusiastic mountaineer Stephen Courtauld (1883-1967), youngest brother of Samuel who later founded the Courtauld Institute of Art. The venue was a private members club, requiring an annual subscription on top of the entrance fees. In 1928 around 500 children from the poorest homes in Westminster’ were allowed to become honorary members’ for one afternoon and to try out the ice with all sorts of sliding devices’. In 1936 the purpose-built premises were re-christened Westminster Ice Rink and opened to the public.