I saw your post when just browsing around and thought I might add a bit if you are still looking. I cannot see what is not clear about what you are looking for!
First of all taking a look at "Indian" history could be an idea. From the pictures you have posted it would appear that Irene was mixed race - often referred to in those days as Anglo-Indians or "Coloureds" - and with the name Gomez most likely of Portuguese descent. This would point to her having been born in Goa which was ruled by the Portuguese who were much more laid-back about mixing with the Indians. This does not mean she may not have moved into British India, where Delhi was, later on. If she was born in Goa finding a record of her birth may be more difficult but it is worth trying the India Office Library, now at the British Library. They have a section which deals with genealogical queries and are very helpful. There is also an organisation called Families in British India (FIBIS) but this may not be of help if you are dealing with Portuguese nationals. I suggest you try googling both India Office Library and FIBIS and see what links look as though they might be useful. Another quirk of history might be a clue as well. For some reason (don't ask me what!) the Indian Railways were almost monopolised by the mixed race/Anglo Indian/coloured community. Senior administrators were mostly, I understand, British, but otherwise it was mixed race folk - and Heaven help any Indians trying to muscle in. As they say in Yorkshire "There's nowt so queer as folk!" If you are still looking and the other leads go nowhere it might be worth trying to find if the Indian Railways have an Archive.
Altogether an intriguing story - the best of luck with your searches.