Does my grandmother have a case so I can pursue legal action after I filed a complaint with the state so they can do their investigation?

You do not explain how the lack of an alarm helped cause her to fall. Speak to some local plaintiffs personal injury attorneys, telling them all the facts. Unless it was a state-owned facility, you do not have to first notify the state. The lawsuit has to be filed by your grandmother as you have no standing to sue on her behalf. Is she mentally competent enough to discuss matters with an attorney?