I don't know, this may sound harsh, but if your daughter is applying for vet school and she was unable to initially contact the vet for an eLOR (you her mother had to make the first contact), which she then followed up on, it sounds like she may need a little more practice in the world before tackling something like vet school. Don't get me wrong, I am a shy person when it comes to talking to authority figures. I always speculate how the people will react, if they like me, if something will change our relationship etc., intern this makes it challenging (scary) for me to contact others, but I still do it.
Anyways, I think if I was a doctor being asked to write a letter by someones mom I would probably refuse too. If the candidate could not swallow their shyness and allow their passion and drive to compel them forward and ask for what they need, then what future do they hold in this field? And still, it is you (her mom) on here seeking out advice on how to attain a good eLOR instead of your daughter. It just makes me wonder where her passion is.
Back on to the subject of the doc in question though, why try and pull teeth from a vet that does not want to write a letter? Your daughter will just end up with a bad letter. Instead of thinking that your daughter wasted 11 months because she did not get an eLOR out of the vet, think of the 11 months of vet experience she gained that she can now add to her application!
I wish your daughter luck in her pursuit.