I was having the exact same issue as others in that my Wii would get stuck while transferring stating that there was an error with the internal memory.
It seems that some of the save files cannot be copied to an SD card and, therefore, the Wii U Transfer Tool will not work.
Here are the steps that worked for me and, maybe, it'll work for you. Note: This solution was adapted from a post by srhode at arstechnica.
1. Put an SD Card into the Wii.
2. Go to Wii Options, Data Management, and then Save Data.
3. Select a save game file. On the screen it will show "Move", "Copy", and "Erase". If the Move and Copy are greyed out, then the save file cannot be moved to an SD Card. For each save file I came across that was like this I hit "Erase".
4. Follow the Wii to Wii U transfer steps as indicated by Nintendo.
Whether or not this is the best solution, I cannot say. But, this solution worked for me. I lost the following save files, which wasn't really a big deal.
MarioKart Wii
TV Show King 2
ExciteBike World Rally
Endless Ocean
SpongeBobSquigglePants
Monster Lab
Elebits
Now, only if the Wii Menu on the Wii U would have access to more memory! Then I could have all of my little Wii downloads on the Wii U.