Legal Aid has nothing whatsoever to do with the payment of judges. It's a system for providing access to the court system for people who can't afford to do it for themselves.
It isn't controlled by local authorities. It's administered by the Legal Aid Agency, which is an agency of central government.
Local authorities have no role in running the court system, which is administered by the Courts and Tribunals Agency (also an agency of the MoJ)
The bulk of police salaries is paid centrally by the Home Office through the Police Fund, which is administered by the local Police & Crime Commissioner. The additional item in your Council Tax bill (technically known as a "precept") is basically a means by which the Commissioner can raise additional funds, subject to limits on his or her ability to increase the precept from year to year.
I can't wait for your private prosecutions against named judges. With your detailed grasp of the operation of the court system, I can't imagine how anything could possibly go wrong.