Can I swap an 830m NVIDIA GPU from an Acer Aspire E5-571g with a 630m NVIDIA GPU from a V3-471g?

There are many factors involved in a GPU swap. The first is that they both have to be using MXM cards of the same type. Without this any other consideration is already void. Secondly, there needs to be space to fit the new card in place of the old one (different card lengths are employed and if the machine doesn't have the space and is positioned such that you can't make the extra space available you are again at a no go).

If both machines employed the same MXM card slot and both cards are the same size you now have to find out what the power draw is. If the sum total of your machine uses 90 watts, 35 of which are the GPU, and the new GPU draws 50 watts on its own, you will need to get an uprated power supply. Whether or not this is possible depends if your model came with an option for a GPU with that much power draw (or more) or whether or not you can use the surrogate's PSU (provided its total power is enough) or get a compatible generic that provides enough.

Failure to have a PSU with enough power to handle the new card will result in performance issues that may make it worse than the original card.

The question that needs answering here, is what about the machine with the better GPU is broken? Perhaps chucking some of the parts of the working machine at the non working one can bring life to it (CPU memory etc depending on what the issue was with the broken machine).