When you say "broke" here, Nathalie, are we talking "split" or "Crushed; mangled; mutilated; destroyed to smithereens" - or something in between?

I got a really good deal on a little 1864 CLOOS fife in "C" ($40) because it was split pretty obviously and would only play about 4 notes, and those poorly.

I have learned that at least on a wooden instrument like a fife, that gobs of glue are not always the way to go when healing splits.
Wood splits when it's natural moisture and resins evaporate, dissipate, and dry out. The cells collapse, causing a cumulative shrinkage in the wood. Something's got to "give", and that is usually a seperation between the grain - ergo, a split.

So after cleaning the accumulated dust and spider-webs off and out of my cutie CLOOS, I merely did what I routinely do in the maintainence of my other fifes; SOPPED it in sesame seed oil, and rubbed as much of it into the grain as it would stand.

Pretty soon,the dry, porous wood (ebony or blackwood, by the looks of it) "drank in" the oil, expanded back to it's original dimensions... and "Violla!" the crack disapeared, and the sweet, clear voice that had been lost for a Century or so came back to life!

Now had I glued the split, and then the fife got wet or properly annointed with oil, the glue would have kept the split from closing and acted like a wedge against the expanding wood, setting up potentially destructive stresses in the thin, delicate wall of the instrument. Not to mention what adulterants like glue can do to harmonic resonance!

Now in the case of that (Peruvian?) pan-pipe; most of the ones I've seen are constructed of a bamboo or reed-like material which is thin but very dense. Shrinkage in a naturally hollow-core stem does not tend to cause splitting as solid-stalked wood will because of the arrangement of the cellular structures, so your super glue might actually be the best option in this case. It seems to work, anyway, and I for one am a big fan of stuff that works!

Given the time, opportunity and inspiration (a really precious commodity among us "Creative" types) I could probably repair or replicate said "Jacobs" flute, but even he (providing that he can be found) might not be able to garuntee that it would sound or play exactly as it did before it's lamentable "accident".

There are so many subtle and mysterious variables in the art of flutecrafting that even the Masters of the art are challenged to make a number of them exactly alike. That's why only a very few of them have, and even fewer still do, make a set of fifes for an entire Corps that will be in tonal and harmonic synchrony.