*is tickled that Kaitlyn thought to mention her*
The first thing that the staff made me think of was Bartitsu. Unfortunately, upon closer examination, it seems like it relies too much on grappling-type combat. However, a combination of two of the French arts that affected the development of Bartitsu may be effective:
canne de combat and
savate. This may be especially effective for your telekinesis nyadacha, as their cobha would allow them to launch themselves more into their opponent.
As for your illusionist nyadacha, I gather that you're looking for something perhaps not entirely as practical, but flashy. For that matter, with the distractions they can create, something slightly less impractical could actually still be quite effective.

For this kind of thing, I would lean toward
Chanquan wushu. I especially love the butterfly kick that this art uses.
For the healing nyadacha, their use of the body's weaknesses made me think of throwing arts, such as
judo, especially its throwing and striking techniques. Though judo also includes grappling, that can be easily circumvented. A good striking technique or a sword thrust following a breath-stealing throw would be effective.
As for humans, you could go anywhere you wanted to, since you don't really have to deal with cobha. You may wish to consider how they are viewed in comparison with the nydacha. If they are considered inferior, and if anything that I've mentioned seems workable, you'd want a simpler art to draw that contrast, such as
karate or
taekwondo. I would lean towards taekwondo myself. Partially because I'm biased, partially because I believe it is more geared toward actual hard-hitting fighting, though that is still dependent on the style. If they are considered superior, I'll have to think about that one some more.

Anyway, I hope my rambling was of some use!

Let me know if you need help deciphering some of the terminology in the articles or if you need any other help at all.
