The original Model K was split into two models: the Model KA and the Model KB. The first got the V8 engine in 1932, while the other got the V12. The later Model KA received a V12 L head engine that shared little in common with its larger counterpart in the KB.
The Lincoln lineup was trimmed down in 1935 during the middle of the Great Depression. The KA and KB were merged into the Model K. Lincoln decided to target the wealthy buyers who could afford a car that cost more than $4,000. This greatly limited their potential sales.
By 1936 it was the 7 passenger Model K model that was getting the most sales with its $4,700 price tag, as few individuals could afford the smaller and slightly less expensive model. The cheaper Lincoln-Zephyr meant that the Model K had little time to live.
Despite the sale of a Sunshine Special Model K convertible limosine to President FDR, the line was discontinued following the introduction of the Continental.
These cars are obviously collectible as vintage classic cars. You can sometimes find one listed on eBay, if you're interested in buying one restored or even a beater to fix up yourself. There also some parts listings on there.