This instructable is about repairing the neck of a guitar. Repairing a broken musical instrument looks like a difficult task, but it can be easy enough. In this case, the neck and finger board broke off cleanly, and the following steps are how I repaired the damage.
Before you begin, make sure the guitar isn't worth more than a few hundred bucks. If it is an expensive instrument, have it repaired properly by a professional. However, if it's a cheap guitar with plywood plates, you could probably by a new one for less than the cost of this repair, so the risk is small.
Other instructables about repairing non-electric guitars:
https://www.instructables.com/id/Guitar-Head-Repair/ (with style)
Step 1: Diagnose
Before you begin the repair, try to figure out what caused the problem. If your kid dropped the guitar, that's easy to know what's the problem and there's nothing you can do about it. On the other hand, the broken neck may be a symptom of another problem.
In the case of my guitar, I found that the previous owner had strung it up with the wrong kind of strings. As you can see from the images below, the guitar was meant for nylon strings. There are structural differences between the two kinds of guitars. The differences are so great that you can feel it simply by lifting the guitar---the steel-string guitar weighs 2-3 times more than a classical guitar. All of that mass is due to the bracing and other measures taken to strengthen a steel-string guitar so that it can resist the pull of the taut strings. When my classical was strung up with steel strings, the neck broke off cleanly. Clearly, fixing the guitar won't solve the problem unless I use nylon strings in the future.
Step 2: Clean
Before gluing the neck back onto the body of the guitar, scrape off the old glue. I used a scraper, a sharp chisel, and a utility knife. Use whatever works for you. Try to minimize the amount of damage to the finish.
Step 3: Apply Glue
Dry fit the pieces together and imagine how you will clamp it so that the glue will dry without disruption.
Apply glue to both the neck and the body. Use your judgment to figure out where the parts will touch. Use lots of glue---the extra will squeeze out, which is easy to deal with. Too little glue will cause voids which weaken the joint. This is much harder to mend, so use plenty of glue.
Step 4: Clamp
Fit the neck in place and clamp it down. I noticed that I had to clamp down the finger board and the heel (touches the side of the body). Turns out the heel doesn't touch if the neck is left to its own devices. I guess the shape counter-acts the tension of the nylon strings.
I used bits of wood to distribute the pressure of the clamps. On the heel, which is round and difficult to clamp, I also used a folded cotton rag to reduce slipping.
Wipe up any excess glue you can reach.
Let the glue dry overnight.
Then, read this instructable---after this repair, you'll have to make adjustments to make the action reasonable.
If you aren't interested in that, you can skip it, string it, and tune your newly repaired guitar: