I struggle with a 2 year old refusing anything to be fair. The little one is unlikely to be making himself sick with anxiety as much as temper. Little ones are super good at knowing what buttons to push
. This type of thing is basically about who is in charge 
The solution is always consistancy which is easy to say after a good nights sleep and not so easy when all you want IS to get some sleep. No giving in , no nights off.
Old fashioned as it sounds its all about routine.
We have done the same thing for 3 of the 4 of ours ( now aged 9,6,4 and nearly 2).
Bedtime countdown, no iPads, no tv, quiet stories , cuddles etc. Into bed, one story, tuck in no fuss. Leave. 2 of our little ones have a nightlight , the other 2 doesn't.
Anyone wakes up and comes out of their rooms then they're taken back to bed, no eye contact ( nothing worth waking up for here ) , tucked back in, simple "night night". And we repeat that with no spoken words for as many times and as long as it takes.
The first 3 have been pretty quick learners. The littlest one has just come out of his bed so next week we'll start the same process with him. We NEVER take them into our bed unless they're ill. The moment you do that -its game over.
I really feel for your daughter but if they don't fix this now they'll be sharing their bed with 2 children pretty soon. I hope they find the way that works for them.