When was the last time a serious attempt was made to fix it that failed? Why did it fail?

Serious attempts are assayed almost every Congress. Often they break down before there's a bill that can get enough votes to pass a floor vote. The last one to pass one chamber was the Border Security, Economic Opportunity, and Immigration Modernization Act of 2013.

They all fail for the same reason. While there is a large majority that would like to change the status quo, there is no majority support for any specific change to the status quo, and there isn't really a compromise that any majority regards as being better than the status quo. The current system is an inelegant kludge, but it's an inelegant kludge that until very recently ended up in a rather stable balance between immigration hardliners and migrant activists and business interests and the various centrist voters.

While all sides criticize it as a broken system, the de facto aspects of the current situation actually may achieve a better and more lasting balance between the policy preferences of those various factions than any legal de jure system could.