It is clear that a wand that has been broken will not work properly and is very hard to mend. The only examples of mended wands are Harry's phoenix wand, which he mended using the Elder wand, and (though it is not explicitly said) Hagrid's wand, which apparently was mended by the Elder wand as well.
However, a wand is nothing more than a tool to channel the magical energy of a wizard. They can be made with different kinds of wood, with various magical cores, and (most important to my point) many shapes and lengths. Coming to think about it, and depending on the way the wand is broken, a single piece of a broken wand seems to be simply a shorter wand of the same wood and core. This is of course a logical and non-canon-supported conclusion. Maybe the wood needs to undergo some kind of process before you install it around the core, and this is damaged when the wand is broken? Maybe there is a missing process in the manufacturing of a wand?
Why wouldn't a broken wand act as a shorter wand?