Question

Why did the United States fail to reach a compromise on slavery? Was there a solution other than war?

Answer 1

The US had many attempts at compromise with varying levels of sucess before the Civil War. However, when Lincoln was elected president (on the platform of stopping the expansion of slavery), the South lost hope and seceded. 
There is always a solution other than war, however, in this case, war had been a looming possibility for quite some time, and compromise probably would've only pushed the war back a year or two.
Hope this helps :)