I did this for my chicken, and had to about once per week for several months until she finally passed away. She was in good spirits so I continued as long as she was still doing her normal activities and seemed happy.
I found that the easiest way to do this was to take an 18 gauge needle, so no syringe. I would poke two holes into her underside (near her vent, a bit below and to the right, and to the left) and put her in a cage in the kitchen. I would put a puppy pee pad in the cage because she would drip a lot of fluid. I also gave her a scrambled egg and pedialyte so she would have some extra nutrients and electrolytes after losing so much fluid. It would take a couple hours to drip out completely. This method was much less stressful for her and me than trying to keep her still while I suctioned the fluid with the syringe. When I was using the syringe in the beginning, I believe I hit an organ because brown fluid came out. Thankfully, she was fine, but I was really scared for that to happen again.
If you want to measure the fluid that comes out, put the needle into her (no syringe) and stand her over the top of a bowl or cup. When the needle is in, the liquid will quickly pour out of her and into the bowl or cup. I also did this every so often just to see how much was coming out (it would be about 2 cups). After I got to know how full her abdomen would feel at 2 cups, I quit measuring it. Truly, it was so easy to just puncture her with the needle, remove the needle, and then let her drip in her cage.
Good luck! Your hen can live a happy life if you just stay on top of keeping her drained.