https://blog.gatunka.com/2009/09/12/using-a-japanese-ime/
It all sounds horribly complicated, although it may be one of those procedures that's easier to do than to describe.
These kind of writing systems really don't seem suited to being input via a conventional keyboard, even with software assistance.
I wonder if things will change [or have changed] with the advent of touch screens and styluses? [stylii?] which would allow the characters to be written manually, using finger or stylus and then have the software convert this to the appropriate typed character?
Typing a Japanese word or phrase is just like typing in English, with the exception of selecting the appropriate kanji from a list (no mouse or stylus required). An experienced typist can do this very quickly, on a physical keyboard or on a smartphone.
For a word like "Japanese" (language), i.e., Nihongo or 日本語, it's actually fewer keystrokes than in English: 5 vs. 8, or 9 if you count the space afterward since Japanese doesn't have the same spacing requirements.
But I'm imagining it from the point of view if I had to learn the Greek or Cyrillic alphabets and type phonetically in them to have the computer then try and guess what I wanted to write in the Latin alphabet. But maybe for Japanese people brought up on this method of entering text, the Latin alphabet is completely familiar, so this disconnect doesn't happen?