What solution did the Kansas-Nebraska Act of 1854 offer to the problem of slavery? What were the consequences of that solution?

Southern states wanted to make sure that the new territories would become slave states and most of the north wanted the opposite result. Douglas and President Franklin Pearce worked out a compromise solution which would let the residents of new states decide by popular vote whether slavery would be allowed. This was the "Popular Sovereignty" provision of the new law. When the Kansas-Nebraska Act was finally passed in 1856, Popular Sovereignty was included.

The act was an opportunity for congress to avoid the slavery question and pass it on to the residents of the new territories. The consequence was disasterous when pro-slavery and anti-slavery settlers poured into Kansas. There were years of brutal fighting and many casualties as a result. Kansas became known as "Bloody Kansas". Ultimately, the anti-slavery settlers prevailed and Kansas was admitted to the union in 1861.

The Kansas-Nebraska Act of 1854, developed by Stephen A. Douglas, created the new territories of Kansas and Nebraska. This law included a provision, known as popular sovereignty, that allowed the people of these territories to decide if slavery would or wouldn’t exist in them. This scenario was different than the last major compromise involving slavery, the Missouri Compromise which banned slavery north of the 36°30’ line except for Missouri.
The popular sovereignty provision had major consequences. Since Nebraska wasn’t suited for slavery, a great deal of attention was focused on Kansas, where slavery could exist successfully. Supporters of both sides of the slavery issue advocated for their cause. In the end, there was a pro-slavery and an anti-slavery legislature elected in Kansas.
As a result of having two legislatures, a great deal of violence occurred in Kansas, which earned it the nickname “Bleeding Kansas.” Pro-slavery forces went to Lawrence and arrested members of the anti-slavery legislature. They also destroyed some property of abolitionists. A few days later, the anti-slavery forces, led by John Brown, got their revenge as they killed several pro-slavery supporters.
When the dust had settled, Kansas eventually entered the country as a free state in 1861. It is doubtful that Stephen A. Douglas, as he developed the Kansas-Nebraska Act, could have envisioned the deadly consequences that the popular sovereignty provision brought to Kansas.