I realize this is an old thread, but to future readers... (regarding the video above)
I suggest not using a small syringe (the needle size is fine, but the container she uses is small) this makes it necessary to poke the hen multiple times.
I had a hen who lived with this for a long while and when she and I drained her, she did not like to be poked therefore "once" is the goal, if possible.
She often had mucky feathers on her bum around the time she needed draining, so I washed her bum before poking, applied some disinfectant (betadine)to the spot I wanted to poke and afterward all the draining is done I washed her bum feathers of the liquid that had messed up her feathers.
Also, if there is enough pressure in the belly the liquid comes out on its own, we just made the hole with the needle and she drained herself onto a towel. The pulling sensation from the syringe is unpleasant no matter the size of it. She made that clear with a sad little trilling sound.
Unfortunately this is palliative care... I have not found any treatment that offers hope.
All the best to your girl (and you as caregiver) this is a sad time, but at least you can spend time together while you spoil her a little at the end of her life. It comes at a time you've likely had a few years together and she is now past laying and enjoying a little retirement. It is a tender time for both of you. I have had three bantam Leghorns (from the same place/same clutch) that had this and I have 2 more who look headed that way. The longest I managed to keep one was about 8 months. I can't make all the pain go, but they still have a quality of life and have a clear desire to live.