I Wonder if This Would Work in the US

In the UK, it is possible to sue the police for not doing their job. Family of murdered woman to sue police and CPS 'for failing to protect her from abusive husband' | Mail Online CPS is the Crown Prosecution Service for you non-Anglophile types.

Police had previously arrested [the husband] twice and ordered him to stay away from his wife, but no charge had been brought against the 36-year-old prior to the fatal attack.

His first day in court was a murder trial. He was convicted. Cold comfort to her family.

I suppose since the UK severely limits your right to defend yourself, trying to force some action from the police makes sense. But this is the worst of all worlds: you can't defend yourself, police do nothing to protect you. (Or next to nothing.)