One may think there are no ways we have not failed God. Now, the question of Christ's crucifixion, I like to think, is more the atoning love and choice of God, rather than a direct result of original sin.
In fact, though seemingly small of a statement, there are many debated theological ideas within your statement. In any case, if you believe in original sin, then the result was death and separation from God, assuming you also take Genisis literally. Now, that said, Christ 'remedied' that situation through His sacrifice.
How have we failed God? Well, could be every way, if that's your focus; however, not everything is a providential test of the Divine. I believe God does allow us to be tested at times, but He's (commonly referred, though transcendent of gender or form, though also existing thus) not mad at you, disappointed, or keeping tabs of your failures. Try to focus more on how you've been obedient, faithful, or who and how you can be with, in, and for God.
Like Paul says, learn to be content and focus on the good rather than the bad. ^^() I know his may not be the response you wanted, but I thought I'd try to chime in more on threads, rather than just reading without responding.