Clean the glass, step back, and the window still looks grey. That's the normal experience of cleaning your own windows in Florida, and it isn't a technique problem. The haze you're seeing is coming from the screen in front of the glass, the sand and grime packed into the track below it, and mineral deposits the glass picked up from a sprinkler.
So the order matters. Screens come out first and get washed and dried properly — a screen that has never been removed holds a surprising amount of dust, pollen, and salt film, and putting a dirty screen back in front of clean glass undoes the whole job. Then the tracks and sills get vacuumed and wiped out, because anything left in there washes back onto the glass the first time it rains. Glass last.
On the glass itself, a squeegee beats a spray bottle and a cloth every time. A cloth moves the dirt around and leaves lint and streaks; a squeegee with a proper solution lifts it off the surface and carries it away in one pass. It's also much faster, which is why nobody who does this professionally uses anything else.
Hard-water spotting is its own problem and the one most people live with unnecessarily. Sprinklers that hit the glass leave mineral deposits that bond to the surface — normal cleaning won't remove them, and a year of them going untreated starts to etch the glass permanently. They need a dedicated mineral treatment, and the real fix is adjusting the sprinkler head so it stops hitting the window at all.
If you're hiring this out, ask whether screens and tracks are included or extra. Plenty of quotes cover glass only, which means you'll get exactly the result you got doing it yourself, just faster. And if your sprinklers have been spraying the same pane for three years, ask for an honest assessment before paying for a treatment — sometimes the etching is past what cleaning can recover.