My wife dropped our D5100 and now the camera will not take pictures and has a black screen. How do you fix this issue?
The 99% likely best answer is, of course, to send it to a competent camera repairer.
Odds are 'It's broken'.
However, just in case, and very very unlikely, try the following.
These have almost no chance of being successful, but in a few cases may work:
Remove & replace lens.
Is mirror stuck up? If so, DO NOT TRY to move it. Take to repairer.
If mirror is NOT stuck up, reattach lens.
Be sure it seats properly.
Try another lens.
It is possible (not likely) that the lens is damaged and that some lens-camera interaction is causing system lockup. At one stage I would have dismissed this as essentially impossible. I've since seen it happen (not with a 5100).
Take out battery.
Leave battery out for one day. NOT 10 minutes. Not 1 hour. A whole 24 hour day.
Reinsert battery and test.
Very very unlikely. A glitching system can (very rarely) lock up its electronics [long explanation with held :-) ] so that it must be powered down and given time for charge to leak away via non-usual paths in the IC(s). Yes it happens. Yes, it's rare. Yes, I've seen this happen. There are ways of inducing this that do not apply here. Chances of it working are slim.
Whats happening is that the shutter stays closed, it is a quite normal issue on D5100, to fix to as follows:
You can check that this is happening by setting camera to video and you will see no screen but sound will be recorded.
Set to manual mode, set minimun shutter speed as so shutter stays open for a while, during this, hold camera with one hand knock it with the other one until shutter un-stucks.
It will work and now you owe me a beer.