When I had to syringe feed my chinchilla, she was more comfortable without a towel. She was harder to handle when she squirmed around, but I think when she saw the towel she would think of me restricting her. It might be best for you to have one though because of the broken leg. Maybe try to test it without a towel if you feel comfortable?
About feeding her with the syringe, I would take the tip of the syringe and put it at the edge of my chinchilla's mouth. She would try to move away , but I tried to keep her head steady. I tried to avoid putting the syringe straight into my chinchillas mouth in case I might have pushed out too much medicine at once, and I didn't want her to breath any of it into her lungs.
As for eating her normal food, my vet recommended softening the pellets in warm water and seeing if my chinchilla would eat some of that. She didn't, but it was an option. Make sure to switch that out often, because it can get messy and will go bad. I eventually had to put my chinchilla on Critical Care because she had stopped eating her food and drinking her water, but after a few days of that she started eating by herself again.
I agree with everything GrayRodent said as well.
