There is another question on here about repairing and its information appears to be out of date. Wikis say you can do this but reddit says you can't, anymore.
It appears it isn't possible to repair pickaxes anymore.
The wiki pages for the pickaxes show that since the Rampaging Koala update, they have been unrepairable. For example, from the Copper Pickaxe page (emphasis mine):
- Perturbed Koala: added
- Rampaging Koala: Reduced durability drastically, made unrepairable, removed Stone Pickaxe from crafting requirement and reduced required Copper Bars, changed description
- Pleased Giraffe: Added price, made uncraftable
(The pages for the other pickaxes say the same thing.)
Additionally, in the Rampaging Koala changelog:
- Removed the ability to repair pickaxes and drills
That update was before 1.0, and there seems to be no evidence to pickaxe repairability having been changed since then.