I have an 01 Mitsubushi Mirage that has been making a clicking noise for about 7 months now.. I stupidly ignored it until today. The tire fell off and I can't tell for sure if it's a broken ball joint or a broken axle. The tire is still connected and the car doesn't lean down on the fender. The tire can still hold it up for support. The axle is hanging down around the middle of the front of the car so I know there's a broken axle but is the ball joint broken too?

 

A broken axle certainly isn't going to cause the wheel to fall off.
Jack it up in normal area so as the suspension in unloaded (hangs down) and you'll see.

 

It sounds like you have a 50/50 shot at getting this right.

 

The clicking for 7 months was probably because your CV boot was torn and the CV joint was getting worn out from debris making its way into the joint and fucking it up. What has probably occured is your CV joint is now broken which would cause what you're describing to happen. Shame because if you catch a torn CV boot early enough you can sometimes just replace the boot, regrease it, and get away with it and save a lot of cash. By the time its clicking or snapping its too late and the axle will need to be replaced before it breaks like yours did. Needs a new axle, possbly more, possibly not.