The Chelsea Physic Gardens say they have a microclimate that supports more tropical plants. I enjoyed the other half of the garden more, where they explain where medicinal plants come from and what they're used for. There's a riverboat pier near there too.
RHS Wisley is also fantastic. Unfortunately slightly more difficult to get to; you really need to drive there.
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.
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.