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So if you Overcharge a wand to use it one extra time per day (p.597), you roll, and on a success it becomes Broken (on a failure it is destroyed). But there's no mention in that section on how you Repair a broken wand, how many hit points it has, or what the DC might be.
You can't use the Mending cantrip because that only works on non-magic objects. So it looks like we need to use the Craft activity called "Repair" (p.243), which restores a number of hit points based on the level of success of your skill roll. Success restores 5hp plus more for level of Training in Craft; crit success restores 10 plus more for level or training.
Per the Repair activity, "The GM sets the DC, but it's usually about the same to Repair a given item as it is to Craft it in the first place." So far, I don't see any rules about specific DCs for making magic items, so I'm not sure if it's based on the minimum level you can be to cast the requisite spells, or if it's based on the "Item Level" per the magic item descriptions? OR p.535 has requirements of Master proficiency in Crafting for items of 9th level or higher (and a wand with a 4th level spell is a 9th level item), and Legendary proficiency for items of 16th level or higher (so wands of 8th or 9th level spells) - so do you also require that proficiency in Crafting to do the Repair???
Page 577 has a table for Material Hardness, Hit Points and Broken Threshold. Unfortunately, wands are not one of the example items. "Thin wood" has 12hp and includes things like Clubs and Chairs and Wooden Shields, all of which are significantly thicker than a wand; so a wand probably has fewer hit points than that (but then again, Magic...) So maybe just a single Success is needed, assuming we can figure out the DC.
For setting DCs, we can try p.503. And here I have to guess the DC scales by level, so I check the Item Level of the wand (p.597, a 4th level spell wand is a 9th level item so a DC 26 on p.503). Or is repairing a wand easy (it's a stick) or very hard (it's MAGIC)?
So that's as far as I can get. If anyone sees something I missed that would clear this up, please share. Also, FAQ candidate. And maybe I'll just introduce Spell-O-Tape...
Just use that 10th level spell that also fixed destroyed wands. Easy!
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Alright let me break it down for you:
It's item DC + rarity of the spell (so it depends on the spell) usually.
So a basic level 3 wooden wand with a common spell (no modifier) is DC 18 to repair or craft. Assuming optimal conditions for repair (so no modifiers), if it is very hard or very easy your gm can adjust the DC.
A basic wooden wand has most likely less than 10 hp, so one successful repair check should be enough to repair it if you are trained (it repairs 10 HP). (Thin wood if you wanted to use as reference lists chair, club, saplings which are sturdier than a basic wooden wand)
As for the higher-level queries of your items, it would be reasonable to require a certain level of proficiency to repair said items. But again as the Book mentions, the GM sets the DC.