How can I convince my colleague to be more organized?
I go by her desk and I see piles of paperwork everywhere to the point she has difficulty finding what she needs. When I try to approach the subject with her, she immediately points out that I need to clean up my own desk (which is true). And on and on.
Even when I clean up my desk and try to approach the topic again, I am dismissed immediately saying she doesn't have time. The manager doesn't really want to get involved in this.
Unless it infringes on your desk space or ability to do work this would be an unprofessional thing to complain about as it endangers your relationship with your coworker in the same way that going to their home and calling it a mess would.
A workspace is personal and as long as it is inoffensive and does not infringe on other's ability to do their job it is not your responsibility.
It is,however worth noting that some employers do have a clear desk policy for security outside of office hours and if this is being infringed that is a different matter. But still not one for you, it is one for her manager.
Do you work in an open office? Having a mess on your desk can be distracting to other people.
(The following applies to my workplace, maybe not yours. Depends on your office culture and might be 'not done' in a lot of cases) If she's distracting you or others I would stack everything up and put that in front of her monitor. That way, the mess is gone directly and she'll be forced to (at least slightly) sort it out.