there's no easy way to do that. I'd start with the area but forget the area text tool - use the area as a guide and manually fill with words. You'll never get the thing happening where letter strokes cross base lines of other lines as per the example if you use area text.
As Roger said, automated solutions usually do not look particularly good. Area text has also the disadvantage that you will have to do extensive formatting to shift baselines and letter spacing and that will fall apart as sson as you change a single letter. I'd do it manually with individual text pieces, even if it seems a drag. It's still the most efficient way.
You may consider a dodgy way of creating the bits of type to adjust and move round:
1) Create Type with more or less the final distribution of words into lines,
2) Save as PDF without preserving editing capabilities and close;
3) Open the PDF and work with the bits.
This may be easier than first placing the individual bits; and you may select groups to move.
It is kind of funny that today everyone thinks that if it is work then it is not worth it. And everyone wants to know what plug in does this thing.
In the old days we would have had a ball piecing it together regardless of how much work and how late we stood up. Actually it would only take a couple of hours but the art director and designer and client would of course analyze it and figure out that that this word should be smaller than that word and that word should should be bigger than that word and it would go on for weeks, of course that will happen in this case as well.
Of course if i were the art director or the designer I would not do anything until two hours before the final deadline so there was no chance to change anything.
Which is a good point to make here do it manually do not rush through it spend small amounts of time on it so it is not a drag and finish it about a half hour before the final deadline and be thankful there is no way to automate this as your client will drive you crazy if there were such a method.
And charge them a lot of money for doing all this manual work.