I’ve always used a squeegee and dishwashing liquid and water in a spray bottle, then blotted off any remaining dribbles with an old handtowel. The residual stains on the mirror may be behind the glass and if so cannot be removed. Its called mirror cancer. Only way to remove those is to replace the mirror. Happens when water/moisture gets into the silvery backing of the mirror. Worth trying with dishwashing spray as above and if that doesn’t fix it it’s probably permanent. Don’t waste your time with windex or similar.