But how did deliberate poisoning with insulin become one of the agreed facts of the case?
There were lots of things Letby agreed (deliberate insulin poisoning, parents' witness statements, colleagues' witness statements, etc) then disputed at the last minute under cross-examination when their significance became clearer to her (and to the jury).
She didn't just agree these things on paper - she also sat there in court while these statements were read out to the jury as agreed facts.
It was only on her cross-examination that she decided to dispute them, once little details on timing, sequence of events, etc were tied together as evidence against her.
The prosecution did point out when she was belatedly disputing these previously agreed details that she clearly had no problem passing notes and interrupting people's testimony when she didn't agree with it at the time it was being said in court, and they pointed out that this begs the question: why is she only disputing this stuff now she's become aware of its significance to the prosecution's case? Why not dispute them earlier when they were being read out to the jury, the way she had done with other things she had mistakenly agreed on paper?
The jury obviously heard all this, and that fact (that she agreed stuff, listened to it being told to them as facts, and then subsequently tried to backtrack at the last minute when she realised those same facts made her look more guilty) would have formed part of their opinion as to whether or not she was guilty.