What’s the best way to make my brass door hardware shiny again?

The problem you’ve got comes from a lacquer coating. It’s intended to prevent tarnishing during storage, but it also prevents polishing in areas where it’s still hanging on. Lacquer thinner, acetone or nail polish remover will probably get rid of the lacquer, but be careful. It might also damage surrounding paint. With the old lacquer gone, use some Brasso metal polish (it works great), then leave it at that. Normal use and an occasional polishing yield a great patina on your door’s brass hardware that just gets better over time.