Your hen is broody, and no rooster is needed for her to go broody (only to make the eggs fertile). If she never saw a rooster in her life she could still become broody.
Broody hens do not lay (they conserve their energy whilst sitting on a clutch).
The cold water treatment is pretty nasty for the hen, and, as you have seen, may not work.Broodiness is due to a hormonal change in the hen and is not down to a choice that she has made.
Either let her have fertile eggs to hatch if you want chicks or separate her from your other hens for a few days with her breast and belly in an environment that they are kept cold (e.g a dog cage on a garage floor). Breaking broodiness can be quite difficult, but has to be done if you don't want chicks.
Sandie