This may be a regional thing, but speaking as an American:

I'd probably say "please put on the kettle". This doesn't make much literal sense -- put what on the kettle? -- but it's a common idiom.

Or, "please start the kettle".

Or more generally: "please make some tea". Sure, this doesn't specify to use a kettle, but that would likely be assumed. As opposed to using what to make tea, a bicycle pump? Oh, okay, you could boil one cup of water in a microwave.

Maybe other phrases. depending on how specific I thought I needed to be.

I don't know of a single word or simple phrase that explicitly expresses the idea of putting water in an electric kettle, plugging it in, and turning it on. Probably because most of the time this is unnecessary detail. If we had an electric kettle and also a kettle that you put on the stove top, would I find it necessary to specify which one I wanted you to use? If I did, then I'd have to spell that out. "Please use the electric kettle to boil some water" or "please make some tea with that cool new electric kettle we just bought" or whatever.

I probably wouldn't say "please boil some water", because you might think I meant in a big pot to make soup or pasta.

If you told me to "set up the kettle", my first thought would be that you had just bought a new kettle and we needed to assemble it. I wouldn't think of pouring water into a kettle and plugging it in as "setting it up".