mini closet - Jack/King stud question

You are showing just jack studs. The king studs and the cripple tie the doorframe to the rest of the framing structure.

Your structure appears to be free standing with nothing tying it to the rest of the wall except the wallboard.

With very gentle handling, it might hold up for a while, but it seems eventually likely to shift and pull free.

Dealing with such a narrow door, I don't think you'll need much more than careful framing, an interior door stop, and don't slam the door so hard you knock the stop loose.

I have a hall closet that's 21" wide inside. The door is framed out with single 2x4 kings and header, with a single cripple on the header, none on the kings. The 1x4 jacks on top and the hinge side, 1x6 jack on the latch side, and 1x1 (3/4" thick, 1" wide) door stop are all nailed to the kings and header about every 24".

I have another 24" wide bedroom closet that has single 2x4 kings and header, no cripples, with the door and door stop screwed directly to them, no jack studs at all. The kings are just nailed to the wall top and bottom plates.

Both of these have held up for 90 yrs so far so I think you can get by with minimal framing as long as the wall isn't load bearing or critical for other support.

If you are tight on space left-right and can't afford the thickness of both jack & king studs you may be able to use a header hanger instead:

They are available from Simpson Strong-Tie & also Mitek in the USA at least.

Since the closet seems to be constructed in free space, it won't have any load-bearing function, so I don't think the lack of double studs is any problem from that point of view.