The latest update for Wii U, dubbed version 5.1.0, will allow you to transfer data from one Wii U console to another. Nintendo is rolling out the update now, so keep an eye out for it!
According to the Big N, both Wii U consoles will need to be present and fully working in order for the transfer to go through. So, if your console it broken, it sadly doesn't look like this method of data recovery will work for you.
I can't see all that many uses for this feature, unless of course Nintendo plans to release a new version of Wii U, in which gamers who want to upgrade their box will have a simple transfer solution that makes the transition a bit less painful. I guess it could also prove handy for those who are upgrading from the standard white model to the deluxe console. But honestly, who bought that silly white version?