"Mr Dan Langham in ‘On Your Toes!’ at the New Hippodrome"
The London Hippodrome
GNU Free Documentation LicenseThe London Hippodrome - Credit: Paul Hermans, Wikimedia Commons

 The London Hippodrome is a real theatre, built in 1900.  Stella Gibbons clearly didn't think much of its structural integrity, if a New Hippodrome was to come into being in 'the near future'.

Hippodrome (lit. "horse stadium) was a common name for theatres and music-halls throughout Britain.

Both Dan Langham and his show appear to be imaginary, but Stella Gibbons may be basing them on performers and productions of the time. She was well-informed about theatre, opera and musical shows, having written theatre reviews for the Lady magazine. Her husband, Allan Bourne Webb (who died in 1959), was an actor and singer, who often appeared on the London stage.