The rogues are mostly non-compos mentis. I can guarantee most of them have been convicted of jack and shit. So killing them would be murder. Yeah anyone in Arkham has never been convicted of a crime. As being there means they're declared no able to stand trial due to mental disorder. I know that's not realistic, most of them wouldn't meet the criteria IRL. But this isn't IRL, and Arkham isn't a prison. It's a mental health institution. There's an entire Batman story where the Joker is found competent to stand trial for a murder, and DA is totally going for the death penalty. But, what a twist, he didn't actually do it. So Batman is running around to find who framed the Joker for murder.
That's the thing he wouldn't have to put one person into office. He'd have to put dozens into office, and keep them there. The Mayor, a majority of the city council, the DA, the police commission. Plus any state and federal reps from Gotham. Bruce would also need his fingers in the Governors office. We're talking like 100's of millions for every election cycle forever. Wayne money stacks ain't that fat. Plus he'd be fighting entrenched interests that like the status quo.