Which is the correct word in the following?
Two of the children borne by her were sent to the orphanage
Two of the children born by her were sent to the orphanage
Which is the correct word in the following?
Two of the children borne by her were sent to the orphanage
Two of the children born by her were sent to the orphanage
The first is correct. See https://www.dictionary.com/browse/borne and scroll down to the 'Usage note'. We write 'The child was born' but 'She has borne a child'.
I found an earlier discussion on the use of "born by" on this site. That was in July 2015. That might help. The Oxford Advanced Learner's Dictionary, sixth edition, has this under the verb "born": "used only in the passive, without by". And there is the following sentence under the verb "bear": "She had borne him six sons". For me the passive is "Six sons had been borne by her", not "born by her". So "borne by" seems to be correct.
Your first sentence "Two of the children borne by her were sent to the orphanage" does not make sense, as the word borne is referred to meaning "as meaning carried, sustained, or endured".
The second sentence "Two of the children born by her were sent to the orphanage" is correct, as the word born refers to being brought in to the world, e.g. a baby being born.
I would maybe even restructure the second sentence to the following: "Two of her children were sent to the orphanage"
Or you could maybe even have it as: "Two of her biological children were sent to the orphanage In this case Biological refers to the children she gave birth to