My next stop has to be Kew Gardens!
After you do Kew, try Sissinghurst.
https://en.wikipedia.org/wiki/Sissinghurst_Castle_Garden?wpr...
Kew also has an extension in Sussex, Wakefield, which is excellent and much less trafficked than the London site.
By the way, one of the best ways to reach Kew is by riverboat from Westminster. Really great way to see London.
Agreed about the riverboats. Tried the Thames clippers for the first time a few weeks ago. They're delightfully empty during weekdays and serve drinks and snacks on board.