Is it appropriate to rewrite and republish another author's useful but very badly written paper?
Personally, I would be very happy to read an abstract along the lines of 


  ABC is an important problem because of XYZ. A solution to this problem was presented already by UWV et al., but is not widely known, partly because of the technical nature of their presentation. Here we give a readable presentation of UVW et al.'s ideas, showing that they can be interpreted in terms of PQR, and we give applications in the field of JKL.


If the paper is mostly a reinterpretation of existing work it might not get into a top journal, but if the problem is important and the solution is a good one, your paper will be a valuable contribution and is likely to be widely cited anyway, since most people implementing it will be working from your paper and will cite it along with the original.

There is no danger of it being seen as plagiarism if you are presenting their ideas in your own words, and state at the start that this is your goal. The other authors will likely be happy as well, because their work will become more widely known as a result of yours. There is no need to involve them directly at all, although of course this is possible as well.