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Jeez, those spokes are WAAAAAY too short. The first few mm of the nipple is just bare shank - no threads at all so you'll be lucky if you have more than a thread or two actually engaged inside the nipple. Longer nipples are not the answer. I wouldn't be surprised if you could yank those spokes out by hand.
New spokes, sorry.
(Edit. I presume you do have some tension in those spokes?)
This amount of threads showing isn't ideal is it? Looks to be about 6/9 threads showing at points. How bad is it and will the wheel explode if ridden?