Vinyl auto-covers are not easy to clean. Mine is so big I never clean it. I do brush or blow debris off it when it's dry.
To actually clean it, you have to do it in sections.
Start with the cover fully open. Close the cover just enough that you have a section sitting on top of the tile at the vault end. Hose off that section with plain water and brush it with your pool brush. If you need to use cleaner, use only non-detergent mild cleaners like diluted Simple Green or Ivory Flakes. Brush again. Flush the soap down the vault drain. Any soap you miss will end up in your pool.
Pull out another section of cover and repeat until done.
Re the SpeedStir, it's not necessary but it does speed the process. When you test as often as TFP-ers do, it is a worthy addition. My recommendation is to make sure you have enough graduated cylinders and enough magnetic beans to run all your tests without having to get up to rinse anything. For me, that means 3 of these items, for FC/CC, CH, and TA tests.