How to repair latch lock that has unscrewed

There's a pivot joint in the middle of the knob shaft which lets the two ends rotate relative to each other. You need to get things lined up so each end of the shaft engages just its side of the lockset. That may take some experimentation to get the amount screwed into each of the knobs just right to balance this out, though the cross-pin should help you find the right position.

Bathroom doors are for courtesy, not security, and are designed to be easily defeated. The solitary purpose of that hole on the end of the knob, and the button behind it, is to unlock the door.

Work that option quite firmly, straight in, with the stoutest thing that'll fit in the hole. If that doesn't work, into the trash it goes.

Since the lock-set is beyond saving, you can poke at the other holes in an effort to release the knob handle from the shaft, and otherwise destructively disassemble it in any manner that won't harm the door or trim. It shouldn't be long before you gain enough access to the mechanism to either spring the lock, or slide the latch. Throw the bits out and off to the big-box to buy a $10 bathroom lockset.